Coordinate geometry. distance between two points. 12a

Transcription

1 Coordinate geometr a Distance etween two points Midpoint of a line segment c Dividing a line segment internall in the ratio a : d Dividing a line segment eternall in the ratio a : e Parallel lines F Perpendicular lines G Applications areas of STud Pthagoras theorem and its application to finding the distance etween two points Calculation of coordinates of the midpoint of a line segment Gradients of parallel and perpendicular lines Finding equations of straight lines (including vertical lines from given information Cutting a line segment internall and eternall in a given ratio Application of coordinate geometr: for eample, design, orienteering, navigation and geometrical proofs a distance etween two points A AC BC B Pthagoras theorem: AB AC + BC ( + ( Hence AB ( + ( The distance etween two points A(, and B(, is: B 0 Quick Questions Coordinate geometr is a ranch of Mathematics with man practical applications. The distance etween two points can e calculated easil using Pthagoras theorem. It is particularl useful when tring to find a distance that is difficult to measure directl. For eample, finding the distance from a point on one side of a lake to a point on the other side. Let A(, and B(, e two points on the Cartesian plane as shown elow. Triangle ABC is a right-angled triangle. A (, B(, C AB ( + ( Chapter Coordinate geometr 9

2 Worked eample Find the distance etween the points A and B in the figure at right. B WriTe From the graph find points A and B. A(, and B(, Let A have coordinates (,. Let (, (, Let B have coordinates (,. Let (, (, Find the length AB appling the formula for the distance etween two points. AB ( + ( [ ( ] + ( ( 6 + ( (correct to decimal places A Worked eample Find the distance etween the points P(, 5 and Q(,. WriTe Let P have coordinates (,. Let (, (, 5 Let Q have coordinates (,. Let (, (, Find the length PQ appling the formula for the distance etween two points. PQ ( + ( [ ( ] + ( 5 ( + ( (correct to decimal places Worked eample Prove that the points A(,, B(, and C(, are the vertices of an isosceles triangle. WriTe Tutorial int-6 Worked eample Plot the points. Note: For triangle ABC to e isosceles, two sides must have the same magnitude. A B From the diagram, AC appears to have the same length as BC. C 0 Maths Quest advanced General Mathematics for the Casio Classpad

3 Find the length AC. AC [ ( ] + [ ( ] ( + ( 0 Find the length BC. BC [ ( ] + [ ( ] ( + ( 0 Find the length AB. AB [ ( ] + [ ( ] ( + ( State our proof. Since AC ΒC, triangle ABC is an isosceles triangle. remember The distance etween two points A(, and B(, is: AB ( + ( eercise a Spreadsheet 0 Distance etween two points distance etween two points We Find the distance etween each pair of points shown at right. We Find the distance etween the following pairs of points. a (, 5, (6, 8 (,, (, c (,, ( 7, 5 d (5,, (0, e (, 5, (, f (,, (5, g (5, 0, ( 8, 0 h (, 7, (, 6 i (a,, (a, j ( a,, (a, We Prove that the points A(0,, B(, and C(, are the vertices of an isosceles triangle. The points P(,, Q(, and R(, are joined to form a triangle. Prove that triangle PQR is equilateral. 5 Prove that the quadrilateral with vertices A(,, B(5,, C(, 0 and D( 5, 0 is a parallelogram. 6 Prove that the triangle with vertices D(5, 6, E(9, and F(5, is a right-angled triangle. 7 The vertices of a quadrilateral are A(,, B(, 8, C(, 9 and D(, 5. a Find the lengths of the sides. Find the lengths of the diagonals. c What tpe of quadrilateral is it? G O 6 K 5 B P C E H A N L F M I J D 5 6 Chapter Coordinate geometr

4 B 8 MC If the distance etween the points (, and ( 5, is 0 units, then the value of is: a 8 c d 0 e 9 MC A rhomus has vertices A(, 6, B(6, 6, C(, and D(,. The coordinates of D are: a (, (, c (, d (, e (, 0 A rectangle has vertices A(, 5, B(0.6, z, C(7.6, 6. and D(,. Find: a the length of CD the length of AD c the length of the diagonal AC d the value of z. Show that the triangle ABC with coordinates A(a, a, B(m, a and C( a, m is isosceles. Midpoint of a line segment We can determine the coordinates of the midpoint of a line segment appling the midpoint formula shown elow. Midpoint formula Consider the line segment connecting the points A(, and B(,. Let P(, e the midpoint of AB. AC is parallel to PD. PC is parallel to BD. AP is parallel to PB (collinear. Hence triangle APC is similar to triangle PBD. But AP PB (since P is the midpoint of AB. Hence, triangle APC is congruent to triangle PBD. Therefore + + Similarl it can e shown that +. In general, the coordinates of the midpoint of a line segment joining the points (, and (, can e found averaging the - and -coordinates of the end points, respectivel. P(, ( D ( A (, ( C (, B(, ( The coordinates of the midpoint of the line segment joining (, and (, are: + +, M, + ( + (, Worked Eample Find the coordinates of the midpoint of the line segment joining (, 5 and (7,. Write Lael the given points (, and (,. Let (, (, 5 and (, (7, Maths Quest Advanced General Mathematics for the Casio ClassPad

5 Find the -coordinate of the midpoint Find the -coordinate of the midpoint Give the coordinates of the midpoint. Hence, the coordinates of the midpoint are (,. 6 Worked Eample 5 The coordinates of the midpoint, M, of the line segment AB are (7,. If the coordinates of A are (,, find the coordinates of B. Write Lael the start of the line segment (, and the midpoint (,. Find the -coordinate of the end point. Let (, (, and (, (7, Find the -coordinate of the end point Give the coordinates of the end point. Hence, the coordinates of the point B are (, 8. 5 Check that the coordinates are feasile. 8 B (, 8 M 7 A (, (7, Chapter Coordinate geometr

6 remember The coordinates of the midpoint of the line segment joining (, and (, are: + +, (, M, + ( + (, eercise B Spreadsheet 075 Midpoint of a segment Midpoint of a line segment We Use the formula method to find the coordinates of the midpoint of the line segment joining the following pairs of points. a ( 5,, (, 8 (,, (, c (0,, (, d (,, (, e (a,, (a, f (a +,, (a, a We5 The coordinates of the midpoint, M, of the line segment AB are (,. If the coordinates of A are (7,, find the coordinates of B. Find: a the coordinates of the centre of a square with vertices A(0, 0, B(,, C(6, and D(, the side length c the length of the diagonals. MC The midpoint of the line segment joining the points (, and (8, is: a (6, (5, c (6, d (, e (5, 5 MC If the midpoint of AB is (, 5 and the coordinates of B are (, 8, then A has coordinates: a (, 6.5 (, c ( 5, d (, e (7, 6 a The vertices of a triangle are A(, 5, B(, and C(,. Find: i the coordinates of P, the midpoint of AC ii the coordinates of Q, the midpoint of AB iii the length of PQ iv the length of BC. Hence show that BC PQ. 7 a A quadrilateral has vertices A(6,, B(,, C(, and D(,. Find: i the midpoint of the diagonal AC ii the midpoint of the diagonal BD. Comment on our finding. 8 a The points A( 5,.5, B(, 0.5 and C( 6, 6 are the vertices of a triangle. Find: i the midpoint, P, of AB ii the length of PC iii the length of AC iv the length of BC. Descrie the triangle. What could PC represent? 9 Find the equation of the straight line that passes through the midpoint of A(, 5 and B(,, and has a gradient of. 0 Find the equation of the straight line that passes through the midpoint of A(, and B(, 5, and has a gradient of. Maths Quest advanced General Mathematics for the Casio Classpad

7 C Dividing a line segment internall in the ratio a : We can also determine the coordinates of a point dividing a line segment internall in a given ratio either plotting the given coordinates and using a first-principles approach or appling a given formula. Worked Eample 6 First-principles method Find the coordinates of the point, P, that divides the line segment joining the points A(, and B(6, internall in the ratio :. Show the end points (, and (, on a sketch graph and show an estimated position of the internal point, P. Write B(6, P A(, 6 Find the -coordinate of P. Since P divides AB in the ratio : then P is located of the length of the line segment AB from A, (P is parts from A and B is parts from A. The -coordinate of P is of the wa etween where and 6. + ( Find the -coordinate of P. Similarl, the -coordinate of P is of the wa etween and. + ( Give the coordinates of the point. Hence, the coordinates of the point P are (5, 9. General formula Consider the line segment connecting the points A(, and B(,. Let P(, e the point on AB that divides it in the ratio a: as shown at right. AC is parallel to PD. PC is parallel to BD. AP is parallel to PB (collinear. Hence, triangle APC is similar to triangle PBD. P(, a A (, ( ( ( C B(, ( D Chapter Coordinate geometr 5

8 Given that AB PB a Note: The ratio a: ma e written in fractional form as a. AC PC then PD AP BD PB a AC ut PD a so ( a( a a a + a + (a + a + a + a+ Similarl it can e shown that a + a+ The coordinates of the point that divides the line segment joining the points (, and (, internall in the ratio a : are: a + a+ a +, a+ Note: When a the formula simplifies to that for the midpoint of a line segment as descried earlier. Worked Eample 7 Formula method Find the coordinates of the point, P, that divides the line segment joining A(, and B(6, internall in the ratio :. Write Lael the end points (, and (,. Let (, (, and (, (6, Find a and. a: : Hence a, a + Use the formula to find the -coordinate and the -coordinate of P. a+ 6 ( + ( a + a+ ( + ( + + Give the coordinates of the point. Hence, the coordinates of the point dividing the line segment in the ratio : are (5, 9. 6 Maths Quest Advanced General Mathematics for the Casio ClassPad

9 Worked eample 8 If P(, is the point that divides the line segment AB internall in the ratio :, find the coordinates of point A if the coordinates of point B are (5, 8. WriTe Tutorial int-6 Worked eample 8 Lael the end point (, and the point P(,. Let (, (5, 8 and (, (, Find a and. a: : Hence a, Find the -coordinate. Let A e (,. Hence, if: a + a+ then 5 ( Find the -coordinate. a + a+ 8 ( Give the coordinates of A. Hence, the coordinates of the point A are (, 0. 6 Check that the coordinates are feasile. 8 0 B 5 P A remember The coordinates of the point P that divides the line segment joining the points (, and (, internall in the ratio a: are: a + a+ a +, a+ B (, a P (, A (, Chapter Coordinate geometr 7

10 eercise C Spreadsheet 0 Dividing a segment internall SkillSHEET. Dividing a line in a given ratio WorkSHEET. dividing a line segment internall in the ratio a : We 6 Use the first-principles method to find the coordinates of the point that divides the line segment joining the following pairs of points internall in the given ratio. a (, 7, (, : (, 7, (, : c (5,, (, : d (,, (6, 9 : We 7 Use the formula method to find the coordinates of the point that divides the line segment joining the following pairs of points internall in the given ratio. a (, 5, (, 5 : (, 5, (, 5 : c (, 8, (7, : d (, 8, (7, : e (, 9, (, 5 5: f (, 9, (, 5 :5 We 8 If P(6, is the point that divides the line segment AB internall in the ratio :5, find the coordinates of A if the coordinates of B are (, 7. The point P(5, divides the line segment joining A(, 5 and B(c, d in the ratio :. Find c and d. 5 MC a The point, P, divides the line segment AB internall in the ratio :. If A is (, and B is (7,, then the coordinates of P are: a (5, (, c (, d (0, 5 e (, 0.5 Point Q(, 0 divides the line segment joining the points A(7, and B(, internall in the ratio: a : : c : d : e : c Points P and Q are the points of trisection of AB in the diagram at right. The coordinates P and Q respectivel are: 5 Q B a (0,, (, (, 0, (, c (0,, (6, d (0,, (, P A 8 e (0,, (5, 6 a Triangle ABC has vertices A(, 5, B(6, 9 and C(, 6. Find: i the coordinates of L, the midpoint of BC ii the coordinates of M, the midpoint of AC iii the coordinates of N, the midpoint of AB. AL, BM and CN are the medians of triangle ABC. A median is a line drawn from the verte of a triangle to the midpoint of the opposite side. Find: i the coordinates of the point on AL that divides it in the ratio : ii the coordinates of the point on BM that divides it in the ratio : iii the coordinates of the point on CN that divides it in the ratio :. c Comment on our findings from i, ii and iii. d The three medians are concurrent. Their common point, usuall laelled G, is called the centroid of the triangle. Graph the triangle ABC and draw the medians AL, BM and CN. Mark the centroid. 7 Triangle PQR has vertices P( 6,, Q(, 9 and R(,. Find: a the midpoints U, V and W of QR, PR and PQ respectivel the coordinates of the centroid, G c the ratio PG:GU and PG:PU. 8 Maths Quest advanced General Mathematics for the Casio Classpad

11 D Dividing a line segment eternall in the ratio a : We can also determine the coordinates of a point dividing a line segment eternall in a given ratio either plotting the given coordinates and using a first principles approach, or appling a given formula. Worked Eample 9 First-principles method Find the coordinates of the point, P, that divides the line segment joining A(, and B(6, eternall in the ratio :. Write Show the end points A(, and B(, on a sketch graph and an estimated position of the eternal point P(,. P(, B(6, Find the -coordinate of P. Since P divides AB eternall in the ratio : then P is located of the length of the line segment AB from A. (P is parts from A and B is parts from A. A(, 6 The -coordinate of P is of the wa etween where and where 6. + ( Find the -coordinate of P. Similarl, the -coordinate of P is of the wa etween and. + ( Give the coordinates of the point. Hence, the coordinates of the point P are (8, 5. 5 Check that the coordinates are feasile. 5 P(8, 5 B(6, A(, 6 8 General formula Consider the line segment connecting the points A(, and B(,. Chapter Coordinate geometr 9

12 Let P(, e an eternal point on the etension of AB that divides it in the ratio a: as shown at right. AD is parallel to BC. PD is parallel to PC (collinear. AP is parallel to BP (collinear. Hence, triangle APD is similar to triangle BPC. Given that AP BP a AD PD then BC AP PC BP a AD ut BC a so a( ( a a a a (a a a a Similarl it can e shown that: a a Worked Eample 0 The coordinates of the point P that divides the line segment joining the points (, and (, eternall in the ratio a: are: a a a, a Formula method Find the coordinates of the point that divides the line segment joining the points (, and (6, eternall in the ratio :. Write Lael the end points A(, and B(,. Let (, (, and (, (6, Find a and. a: : Hence a, a a Use the formula to find the -coordinate and the -coordinate of the required point. a a 6 ( ( ( ( Give the coordinates of the point. Hence, the coordinates of the point dividing the line eternall in the ratio : are (8, 5. A (, a ( B, a A(, P(, C D B(, P(, 0 Maths Quest Advanced General Mathematics for the Casio ClassPad

13 Worked eample If P(, is the point that divides the line segment AB eternall in the ratio :, find the coordinates of A if the coordinates of B are (5, 8. WriTe Tutorial int-6 Worked eample Lael the end point (, and the point P(,. Let (, (5, 8 and (, (, Find a and. a: : Hence a, Find the -coordinate. Let A e (,. Hence, if: a a then 5 ( a Find the -coordinate. a 8 ( Give the coordinates of A. Hence, the coordinates of the point A are (,. remember The coordinates of the point, P, that divides the line segment joining the points (, and (, eternall in the ratio a: are: a a a, a P(, a B(, A(, Chapter Coordinate geometr

14 eercise d Spreadsheet 0 Dividing a segment eternall e dividing a line segment eternall in the ratio a : We 9 Use the first-principles method to find the coordinates of the point that divides the line segment joining the following pairs of points eternall in the given ratio. a (, 7, (, : (, 7, (, : c (, 5, (, 5 : d (, 5, (, 5 : We 0 Use the formula method to find the coordinates of the point that divides the line segment joining the following pairs of points eternall in the given ratio. a (, 8, (7, : (, 8, (7, : c (, 9, (, 5 5: d (, 9, (, 5 :5 e (5,, (, : f (5,, (, : We The point P(6, is the point that divides the line segment AB eternall in the ratio :5. Find the coordinates of A if the coordinates of B are (, 7. The point P(5, divides the line segment joining A(, 5 and B(c, d eternall in the ratio :. Find c and d. 5 MC P is the point that divides the line segment AB eternall in the ratio :. If A is (, and B is (7, then the coordinates of P are: a (0, 5 ( 5, 5 c (8, d (, e (6, 6 MC Point Q( 5, divides the line segment joining the points A(7, and B(, eternall in the ratio: a : : c : d : e : 7 A give wa sign has the shape of an equilateral triangle with 87 cm a side length of 87 cm. The sign is attached in two places to a metal pole. a How far from the top of the sign should the holes e drilled if the top hole divides the vertical height of the sign in the ratio :9 and the ottom hole in the ratio 8:9? How high is the top of the sign from the ground if the distance to the ase of the pole from the top and ottom of the sign is in the ratio 7:6? parallel lines In a previous chapter, Linear and non-linear graphs (Chapter 0, we investigated linear graphs and equations. We are now going to investigate further properties of straight lines. The equation of a straight line ma e epressed in the form: m + c where m is the gradient of the line, and c is the -intercept. The gradient can e calculated if two points, (, and (, are given. m An alternative form for the equation of a straight line is: a + + c 0 where a, and c are constants. Another alternative form is: m( where m is the gradient and (, is a point on the line. Maths Quest advanced General Mathematics for the Casio Classpad

15 Worked eample Show that AB is parallel to CD given that A has coordinates (, 5, B has coordinates (5, 7, C has coordinates (,, and D has coordinates (, 5. WriTe Find the gradient of AB. Let A(, 5 (, and B(5, 7 (, Since m 7 m AB ( 5 5 ( 6 Find the gradient of CD. Let C(, (, and D(, 5 (, 5 m CD ( Compare the gradients to determine if the are parallel. (Note: means is parallel to. 7 Since parallel lines have the same gradient and m AB m CD, then AB CD. Collinear points lie on the same straight line. Worked eample Show that the points A(, 0, B(, and C(0, are collinear. WriTe Find the gradient of AB. Let A(, 0 (, and B(, (, Since m m AB 0 Tutorial int-65 Worked eample Find the gradient of BC. Let B(, (, and C(0, (, m BC 0 6 Show that A, B and C are collinear. Since m AB m BC then AB BC Since B is common to oth line segments, A, B and C must lie on the same straight line. That is, A, B and C are collinear. Chapter Coordinate geometr

16 Worked eample Find the equation of the straight line that passes through the point (, 6 and is parallel to the line +. In order to find the equation of a straight line, we need to know the gradient and a point on the line. One point is given and since the line is parallel to +, the gradients will e the same. Use the formula m( and sustitute the coordinates of the point and the gradient to find the equation of the line. WriTe Point on the line: (, 5 Gradient: m. m( 5 ( 5 6 remember. The equation of a straight line ma e epressed in the form: m + c where m is the gradient of the line and c is the -intercept, or m( where m is the gradient and (, is a point on the line.. The gradient can e calculated if two points, (, and (, are given using m. Parallel lines have the same gradient.. Collinear points lie on the same straight line. eercise e Spreadsheet 06 Gradient SkillSHEET. Epressing the equation of a straight line in the form m + c parallel lines We Find if AB is parallel to CD given the following coordinates. a A(,, B(, 9, C(0, 0, D(5, 0. A(,, B(8,, C( 6,, D(, 6. c A(, 0, B(,, C(, 0, D(8, 6. d A(,, B(,, C(, 0, D(, 5. e A(, 0, B(, 5, C(, 5, D(7, 5. f A(, 6, B( 5, 0, C(0, 0, D(5,. Which pairs of the following straight lines are parallel? a c d + e f 6 0 g + h 5 We Show that the points A(0,, B(5, and C( 5, 5 are collinear. Maths Quest advanced General Mathematics for the Casio Classpad

17 Spreadsheet 085 Parallel checker WorkSHEET. F Show that the line that passes through the points (, 9 and (0, also passes through the point (6, 6. 5 In each of the following, show that ABCD is a parallelogram. a A(, 0, B(,, C(,, D(0, A(,, B(0,, C(,, D(0, c A(.5,.5, B(0,, C(.5,.5, D( 5, 5 6 In each of the following, show that ABCD is a trapezium. a A(0, 6, B(,, C(0,, D( 5, 9 A(6,, B(8, 6, C(,, D(, c A(, 7, B(,, C( 0.6,.6, D(, 7 MC The line that passes through the points (0, 6 and (7, 8 also passes through: a (, (5, c (, 0 d (, 8 e (, 8 MC The point (, 5 lies on a line parallel to Another point on the same line as (, 5 is: a (, 9 (, c (, 0 d (, e (, 9 We Find the equation of the straight line given the following conditions: a passes through the point (, and parallel to + 5. passes through the point (, and parallel to +. perpendicular lines In this section, we shall eamine some of the properties of perpendicular lines. Oserving the graphs can e ver useful in investigating these properties. Consider the diagram elow, where the line segment AB is perpendicular to the line segment BC. Line AC is parallel to the -ais. Line BD is the height of the resulting triangle ABC. Let m AB m B a tan(θ α a θ Let m BC m a c tan(α a Hence m m m or m m Hence, if two lines are perpendicular to each other, then the product of their gradients is. Two lines are perpendicular if and onl if: m m or m m A θ D c α C Chapter Coordinate geometr 5

18 Worked eample 5 Show that the lines 5 + and are perpendicular to one another. Tutorial int-66 Worked eample 5 WriTe Find the gradient of equation. 5 + Hence m 5 Find the gradient of equation Rewrite in the form m + c Hence m 5 Test for perpendicularit. (The two lines are perpendicular if the product of their gradients is. m m 5 5 Hence, the two lines are perpendicular to each other. remember Two lines are perpendicular if and onl if: m m or m m. eercise F Spreadsheet 085 Perpendicular checker perpendicular lines We 5 Show that the lines 6 and are perpendicular to one another. Determine if AB is perpendicular to CD, given the following coordinates. a A(, 6, B(, 8, C(, 6, D(, A(,, B(, 9, C(0,, D(7, c A(,, B(, 8, C( 5,, D(5, 0 d A(, 5, B(0, 0, C(5,, D( 0, 8 e A(, 9, B(, 6, C( 5, 8, D(0, f A(,, B( 8, 5, C( 6,, D(, Determine which pairs of the following straight lines are perpendicular. a c d + e + f g + 6 h + 0 Show that the following sets of points form the vertices of a right-angled triangle. a A(,, B(,, C(, 7 A(,, B(,, C(, c A(0, 5, B(9,, C(, 5 Prove that the quadrilateral ABCD is a rectangle when A is (, 5, B(6,, C(, and D(,. 6 Maths Quest advanced General Mathematics for the Casio Classpad

19 6 Find the equation of the straight line that cuts the -ais at and is perpendicular to the equation 6. 7 Calculate the value of m for which the following pairs of equations are perpendicular to each other. a 5 7 and + m and 5 + m 8 Prove that the quadrilateral ABCD is a rhomus, given A(,, B(, 5, C(5, 6 and D(,. Hint: The diagonals of a rhomus intersect at right angles. 9 MC The gradient of the line perpendicular to the line with equation 6 is: a 6 c d e 0 MC Triangle ABC has a right angle at B. The vertices are A(, 9, B(, 8 and C(, z. The value of z is: a 8 c d 7 e G applications In this section we look at two important applications: the equation of a straight line, and equations of horizontal and vertical lines. The equation of a straight line The equation of a straight line can e determined two methods. Interactivit int-0979 Applications of coordinate geometr The m + c method requires the gradient, m, and a given point to e known, in order to estalish the value of c. Note: Since the value of c represents the -intercept, it can e sustituted directl if known. Worked eample 6 Find the equation of the straight line that passes through the point (, and is parallel to the straight line with equation +. WriTe Write the general equation. m + c Find the gradient of the given line. + has a gradient of Hence m Sustitute for m in the general equation. so + c Sustitute the given point to find c. (, (, ( + c 6 + c c 7 5 Sustitute for c in the general equation. 7 or 7 0 The alternative method comes from the gradient definition. m Hence m( Chapter Coordinate geometr 7

20 Using the general point (, instead of the specific point (, gives the general equation: m( This requires the gradient, m, and a given point (, to e known. Worked Eample 7 Find the equation of the line that passes through the point (0, and is perpendicular to a straight line with a gradient of 5. Write Find the gradient of the perpendicular line. Given m 5 m 5 Sustitute for m and (, in the general equation. Since m( and (, (0, then ( ( Horizontal and vertical lines For horizontal lines the gradient is equal to zero, and so the equation m + c ecomes c. Notice that does not appear in the equation ecause there is no -intercept. Horizontal lines are parallel with the -ais. In the case of vertical lines, the gradient is infinite or undefined. The general equation for a vertical line is given a. In this case, just as the equation suggests, a represents the -intercept. Notice that does not appear in the equation ecause there is no -intercept. Vertical lines are parallel with the -ais. The graphs of and are shown elow to highlight this information. 0 0 Worked Eample 8 Find the equation of: a the vertical line that passes through the point (, the horizontal line that passes through the point (, 6. a For a vertical line, there is no -intercept so does not appear in the equation. The -coordinate of the point is. For a horizontal line, there is no -intercept so does not appear in the equation. The -coordinate of the point is 6. Write a 6 8 Maths Quest Advanced General Mathematics for the Casio ClassPad

21 Worked Eample 9 Find the equation of the perpendicular isector of the line joining the points (0, and (6, 5. Find the gradient of the line joining the given points using the general equation. Write Let (0, (, Let (6, 5 (, m 5 ( m Find the gradient of the perpendicular line. For lines to e perpendicular, m Find the midpoint of the line joining the given points. m m Hence (, are the coordinates of the midpoint. Sustitute for m and (, in the general equation. 5 Simplif removing the fractions. (a Multipl oth sides. ( Multipl oth sides. Note: The diagram at right shows the geometric situation. Since m( and (, (, and m then ( ( ( (6, 5 6 Chapter Coordinate geometr 9

22 Worked eample 0 ABCD is a parallelogram. The coordinates of A, B and C are (, 5, (, and (, respectivel. Find: a the equation of AD the equation of DC c the coordinates of D. WriTe a Draw the parallelogram ABCD. Note: The order of the lettering of the geometric shape determines the links in the diagram. For eample: ABCD means join A to B to C to D to A. This avoids an amiguit. a 5 A D B C Find the gradient of BC. m BC State the gradient of AD. Since m BC and AD BC then m AD Using the given coordinates of A and the gradient of AD find the equation of AD. Find the gradient of AB. m AB State the gradient of DC. Since m AB and DC AB then m DC Using the given coordinates of C and the gradient of DC find the equation of DC. c Solve simultaneousl to find D, the point of intersection of the equations AD and DC. Note: Alternativel, a CAS calculator could e used to determine the point of intersection of AD. Tutorial int-67 Worked eample 0 + c Let (, (, 5 5 ( + c c Hence, the equation of AD is c Let (, (, ( + c c 0 Hence, the equation of DC is. c Equation of AD: + [] Equation of DC: [] [] []: 0 + Sustituting in []: ( Hence, the coordinates of D are (,. 50 Maths Quest advanced General Mathematics for the Casio Classpad

23 remember The equation of a straight line can e determined two methods:. The m + c method. This requires the gradient, m, and a given point to e known, in order to estalish the value of c. If the -intercept is known, then this can e directl sustituted for c.. Alternative method: m( This requires the gradient, m, and a given point (, to e known.. The general equation for a vertical line is given a and a horizontal line is given c. eercise G Spreadsheet 09 Equation of a straight line applications We 6 Find the equation of the straight line that passes through the point (, and is parallel to the straight line with equation 5. We 7 Find the equation of the line that passes through the point (, 7 and is perpendicular to a line with a gradient of. Find the equations of the following straight lines. a Gradient and passing through the point (, 5. Gradient and passing through the point (,. c Passing through the points (, and (,. d Passing through the points (, and (6, 5. e Passing through the point (5, and parallel to f Passing through the point (, 6 and parallel to 0. g Passing through the point (, 5 and perpendicular to Find the equation of the line which passes through the point (, and is: a parallel to the straight line with equation 0 perpendicular to the straight line with equation 0. 5 Find the equation of the line that contains the point (, and is: a parallel to the straight line with equation 5 0 perpendicular to the straight line with equation We 8 Find the equation of: a the vertical line that passes through the point (, 8 the horizontal line that passes through the point ( 5, 7. 7 MC a The vertical line passing through the point (, is given : a c d + e Which of the following points does the horizontal line given the equation 5 pass through? a ( 5, (, 5 c (, 5 d (5, e (5, 5 c Which of the following statements is true? a Vertical lines have a gradient of zero. The -coordinates of all points on a vertical line are the same. c Horizontal lines have an undefined gradient. d The -coordinates of all points on a vertical line are the same. e A horizontal line has the general equation a. Chapter Coordinate geometr 5

24 d Which of the following statements is false? A Horizontal lines have a gradient of zero. B The straight line joining the points (, and ( 7, is vertical. C Vertical lines have an undefined gradient. D The straight line joining the points (, and ( 7, is horizontal. E A horizontal line has the general equation c. 8 The triangle ABC has vertices A(9,, B(, 6 and C(,. a Find the midpoint, M, of BC. Find the gradient of BC. c Show that AM is the perpendicular isector of BC. d Descrie triangle ABC. 9 WE 9 Find the equation of the perpendicular isector of the line joining the points (, and ( 5,. 0 Find the equation of the perpendicular isector of the line joining the points (, 9 and (, 0. WE 0 ABCD is a parallelogram. The coordinates of A, B and C are (,, (, and (, respectivel. Find: a the equation of AD the equation of DC c the coordinates of D. The map shows the proposed course for a acht race. Buos have een positioned at A(, 5, B(8, 8 and C(, 6, ut the last uo s placement, D(0, w, is et to e finalised. a How far is the first stage of the race, that is, from the start, O, to uo A? The race marshall oat, M, is situated halfwa etween uos A and C. What are the coordinates of the oat? c Stage of the race (from C to D is perpendicular to stage (from B to C. What is the gradient of CD? d Find the linear equation that descries stage. e Hence determine the eact position of uo D. f An emergenc oat is to e placed at point E, of the wa from uo A to uo D. Into what internal ratio does point E divide the distance from A to D? g Determine the coordinates of the emergenc oat. h How far is the emergenc oat from the hospital, located at H, km North of the start? MC a The equation of the line passing through the point (, and parallel to the line + 0 is: A + 0 B 5 0 C 0 0 D 0 E The equation of the perpendicular isector of the line segment AB where A is (, 5 and B is (, 7 is: A + B 8 C + D + E c The coordinates of the centroid of triangle ABC with vertices A(, 8, B(9, 6 and C(, are: A (, 5 B (0, 6 C (, 6 D (5, 7 E (, 7 To suppl cities with water when the source is a long distance awa, artificial channels, called aqueducts, ma e uilt. More than 000 ears after it was uilt, a Roman aqueduct still stands in southern France. It rought water from a source in Uzès to the cit of Nîmes. The aqueduct does not follow a direct route etween these two locations as there is a mountain range Buo A H Scale: unit km M Buo B Buo D Buo C O (Start E N 5 Maths Quest Advanced General Mathematics for the Casio ClassPad

25 etween them. The tale shows the approimate distance from Uzès along the aqueduct to each town (or in the case of Pont du Gard, a ridge and the aqueduct s height aove sea level at each location. Distance from Height of aqueduct aove Location Uzès (km sea level (m Uzès 0 76 Pont du Gard (ridge 6 65 St. Bonnet 5 6 St. Gervas Nîmes a Show the information in the tale as a graph with the distance from Uzès along the horizontal ais. Join the plotted points with straight lines. Calculate the gradient of the steepest part of the aqueduct (in m/km. c Suppose the aqueduct started at Uzès and ended at Nîmes, ut had a constant gradient. Write a linear equation to descrie its course. d Using the equation found in part, calculate the height of the aqueduct at the Pont du Gard. This calculated height is higher than the actual height. How much higher? e Wh do ou think the Romans made the first part of the aqueduct steeper than the rest? Chapter Coordinate geometr 5

26 Summar Distance etween two points The distance etween two points A(, and B(, is: Midpoint of a line segment AB ( + ( The coordinates of the midpoint of the line segment joining (, and (, are: + +, (, M, + ( + (, Dividing a line segment internall in the ratio a: The coordinates of the point that divides the line segment joining the points (, and (, internall in the ratio a: are: a + a+ a +, a+ B(, a P(, A(, Dividing a line segment eternall in the ratio a: The coordinates of the point that divides the line segment joining the points (, and (, eternall in the ratio a: are: a a a, a P(, a B(, A(, Parallel lines Parallel lines have the same gradient. Collinear points lie on the same straight line. Perpendicular lines Two lines are perpendicular if and onl if: m m or m m Applications Equations of a straight line:. Gradient and -intercept form: m + c where m (the gradient and c is the -intercept. General form: a + + c 0 5 Maths Quest Advanced General Mathematics for the Casio ClassPad

27 To find the equation of a straight line:. Given gradient and -intercept m + c. Given gradient and a point m( or m + c method. Given two points Find m, then use: m( or m + c method The general equation for a vertical line is given a and a horizontal line is given c. Chapter Coordinate geometr 55

28 chapter review Short answer Find the distance etween the points (, and (7,. Prove that triangle ABC is isosceles given A(,, B(, 7 and C(,. Show that the points A(,, B(, and C(8, 0 are the vertices of a right-angled triangle. The midpoint of the line segment AB is (6,. If B has coordinates (, 0, find the coordinates of A. 5 Find the coordinates of the point which divides the line joining the point A(, 6 and the point B(, internall in the ratio :. 6 Find the coordinates of the point which divides the line joining the points (, 8 and (5, eternall in the ratio :. 7 Show that the points A(,, B(5, and C(, 5 are collinear. 8 Show that the lines and are perpendicular to one another. 9 Find the equation of the straight line passing through the point (6, and parallel to the line + 0. Eam tip To find the straight line ou need a point on the line and its gradient. 0 Find the equation of the line perpendicular to and having the same -intercept. Find the equation of the perpendicular isector of the line joining the points (, 7 and (,. Find the equation of the straight line joining the point (, 5 and the point of intersection of the straight lines with equations and + 5. Using the information given in the diagram. a Find: i the gradient of AD B(, 9 ii the gradient of AB 9 iii the equation of BC C iv the equation of DC A v the coordinates of C. O D Descrie quadrilateral 5 9 ABCD. In triangle ABC, A is (, 5, B is (, and C is (8,. a Find: i the gradient of BC ii the midpoint, P, of AB iii the midpoint, Q, of AC. Hence show that: i PQ is parallel to BC ii PQ is half the length of BC. 5 Triangle ABC has vertices A(a,, B(, 6 and C(5,. The centroid, G, of the triangle has coordinates (,. a Find: i the midpoint, M, of BC ii the coordinates of A iii the gradient of BC iv the gradient of AM v the length of AB vi the length of AC. Descrie triangle ABC. Multiple choice The distance etween the points (, 5 and (6, 7 is: A 5 B 9 C D 9 E The midpoint of the line segment joining the points (, and (, 7 is: A (, 5 B (, 0 C ( 6, D (, E (, If the midpoint of the line segment joining the points A(, 7 and B(, has coordinates (6,, then the coordinates of B are: A (5, B (0, 6 C (9, D (.5,.5 E ( 9, C is a point that divides the line segment AB internall in the ratio :. If A is the point (, and B is the point (, 5, then the coordinates of C are: A (0, B (, C (, D (, E (, The following information refers to questions 5, 6, 7 and 8. Triangle ABC has vertices A(, 5, B(, and C( 6,. 56 Maths Quest Advanced General Mathematics for the Casio ClassPad

29 5 The median from A meets the line segment, BC, at M. The coordinates of M are: A (, B (, C (, D (, E (0, 6 The centroid divides a median in the ratio :. The coordinates of the centroid are: A (0, B (, C (0, D ( E (, 7 The gradient of the median, AM, is: A B 7 C undefined D E 8 The equation of the median, AM, is: A 7 0 B C 7 0 D + 0 E D is a point that divides the line segment AB eternall in the ratio :. If A is the point (, and B is the point (7, 6, then the coordinates of D are: A ( 8, B (8, C (, 5 D (7, E (8, 0 The gradient of the line joining the points (, 7 and (5, 8 is: A 5 B C 5 D 7 E If the points ( 6,, (, and (, are collinear, then the value of is: A B. C D 5 6 E The gradient of the line perpendicular to is: A B D E C The equation of the line perpendicular to + 0 and passing through the point (, is: A B + 0 C D E 0 The following information refers to questions, 5, 6 and 7. The diagram at right shows a square inscried in a circle. (a, The square has D A (, coordinates A(,, B(,, C( 5, - and D(a,. C B ( 5, (, The circle has a radius of: A 0 units B 7.07 units C 6 units D 5 units E units 5 The coordinates of the centre are: A (, 0 B (, 0 C (0, D (, E (0, 6 The gradient of the diagonal, BD, is: A B C D 5 E 7 The coordinates of the point D are: A (, 6 B (, 6 C ( 6, D (, E ( 6, Etended response ABCD is a quadrilateral with vertices A(, 9, B(7,, C(, and D(a, 0. Given that the diagonals are perpendicular to each other, find: a the equation of the diagonal AC the equation of the diagonal BD B c the value of a. The centroid, G, of a triangle ABC divides the medians internall in the ratio :. For eample: F D G AG:GD : where D is the midpoint of BC. A B C is a triangle with coordinates A (5,, B (, 5 and C (6, 9. Find the coordinates of the centroid, G. A E C Chapter Coordinate geometr 57

30 The course for a car rall is planned so that each participating team must pass four checkpoints in order to complete the course at the point where it first egins. The first checkpoint, A, is located 8 km south and 5 km east of the start, while the third checkpoint, C, is 5 km north of A and 8 km east of the start. a Find the distance to checkpoint A. Find the coordinates of checkpoint C. c Checkpoint B is located of the wa from A to C. Find the coordinates of B. d Find the distance from A to C. e Checkpoint D is located m km directl east of the start. One of the teams realises that their car is ver low on fuel and decides to drive directl from B to a service station at D. The know that this will save them 5.0 km of travel. Write an epression for the distance from B to D. f Write an epression for the distance from B to D travelling via checkpoint C. g Find the coordinates of checkpoint D. h Find the total distance of the course (without taking an shortcut. An architect decides to design a uilding with a -metre-square ase such that the eternal walls are initiall vertical to a height of 50 metres, ut taper so that their separation is 8 metres at its peak height of 90 metres. A profile of the uilding is shown with the point (0, 0 marked as a reference at the centre of the ase. a Write the equation of the vertical line connecting A and B. Write the coordinates of B and C. c Find the length of the tapered section of wall from B to C. d The top floor of the uilding is on a level with point T that divides BC internall in the ratio 9:. Find the height of the top floor of the uilding. 5 In a game of lawn owls, the oject is to owl a iased all so that it gets as close as possile to a smaller white all called a jack. During a game, a plaer will sometimes owl a all quite quickl so that it travels in a straight line in order to displace an opponents guard alls. In a particular game, plaer X has guard alls close to the jack. The coordinates of the jack are (0, 0 and the coordinates of the guard alls are A(, 5 and B(, Plaer Y owls a all so that it travels in North a straight line toward the jack. The all is owled from the position S, with the coordinates ( 0,. 0 a Will plaer Y displace one of the guard alls? If so, which one? Due to ias, the displaced guard all is knocked so that it egins to travel in a straight line (at right angles to the path found in part a. Find the equation of the line of the guard all. c Show that guard all A is initiall heading directl toward guard all B. d Given its initial velocit, guard all A can travel in a straight line for metre efore its ias affects it path. Calculate and eplain whether guard all A will collide with guard all B. Start 8 B C 5 8 A B 50 m A S( 0, C T 8 m 0 m B( 57, 5 A(, (Not to scale 0 D m 90 m Test Yourself Chapter 58 Maths Quest advanced General Mathematics for the Casio Classpad

31 activities chapter opener 0 Quick Questions: Warm up with ten quick questions on coordinate geometr. (page 9 a distance etween two points Tutorial We int-6: Watch how to prove that three points are vertices of an isosceles triangle. (page 0 Spreadsheet 0: Investigate the distance etween two points. (page Midpoint of a line segment Spreadsheet 075: Investigate the midpoint of a segment. (page c dividing a line segment internall in the ratio a: Tutorial We8 int-6: Watch how to find the coordinates of a point of a line segment given it is divided internall in a ratio :. (page 7 s Spreadsheet 0: Investigate dividing a segment internall. (page 8 SkillSHEET.: Practise dividing a line in a given ratio. (page 8 WorkSHEET.: Solve prolems of distance etween two points, locate the coordinates of the midpoint of a segment and determine coordinates of the segment internall dividing a line using first principles and ratios. (page 8 d dividing a line segment eternall in the ratio a: Tutorial We int-6: Watch how to find the coordinates of a point of a line segment given it is divided eternall in a ratio :. (page Spreadsheet 0: Investigate dividing a segment eternall. (page e Parallel lines Tutorial We int-65: Watch how to show that three points are collinear. (page s Spreadsheet 06: Investigate gradients. (page SkillSHEET.: Practise epressing the equation of a straight line in the form m + c. (page Spreadsheet 085: Investigate parallel lines. (page 5 WorkSHEET.: Solve more comple prolems of distance etween two points, locate the coordinates of the midpoint of a segment, determine coordinates of the segment internall and eternall dividing a line using ratios and demonstrate understanding of parallel lines and collinear points. (page 5 F Perpendicular lines Tutorial We5 int-66: Watch how to show that two straight lines are perpendicular. (page 6 Spreadsheet 085: Investigate perpendicular lines. (page 6 G applications Interactivit Applications of coordinate geometr int-0979: Appl our knowledge of coordinate geometr using the interactivit. (page 7 Tutorial We0 int-67: Watch how to determine equations of two sides of a parallelogram and find the coordinates of a verte. (page 50 Spreadsheet 09: Investigate the equation of a straight line. (page 5 chapter review Test Yourself: Take the end-of-chapter test to test our progress. (page 58 To access ebookplus activities, log on to Chapter Coordinate geometr 59

32 EXAM PRACTICE Chapters 8 TO SHORT ANSWER 0 minutes π Convert cis to eact Cartesian coordinates. marks Prove induction that n (n + is alwas divisile for all integer values of n. marks The partial fraction can e resolved as follows: + A B C + ( ( + a + + ( a + a Determine the values of a and. marks Show that C - 7. marks c Hence, determine the values of A and B. marks MULTIPLE CHOICe 0 minutes Each question is worth mark. When converted to polar form the circle with equation ( - + ( + 0 would ecome which one of the following: A r 8 B r ( cos (θ - sin (θ C r (sin (θ - cos (θ 0 D r sin( θ + cos ( θ + 0 E r cos( θ + sin ( θ A perpendicular line PQ is drawn so that is cuts the line segment AB in the ratio of :. The coordinates of A and B are respectivel (, and (5,. The equation PQ would e: A - B + 7 C + D + E + Epressed in the form + i the polar equation z cis 5π 6 would e which one of the following? A i B i C + i D i E i 5 p q r Truth value F T T T F T F F T F F F T F T F Which one of the following compound statements represents the truth value for the truth tale aove? A (p q r B p (r q C (p q r D (p r q E p (r q 6 O 6 The graph aove shows the feasile region of a set of linear inequalities. If the ojective function z - a is maimised at the point (, then the value of a could e: A - B - c 0 D E 6 The line - + intersects the paraola ( + - ( + 0 at points P and Q. If the coordinates of P are (a, then the values of a and would e respectivel: A B C D E and ( and ( and ( 7 5 and ( 7 5 ( and ( Maths Quest Advanced General Mathematics for the Casio ClassPad

33 etended response 5 minutes The shape of a shade sail can e modelled using part of a rectangular hperola and two linear equations. a If the equation of the rectangular hperola is given 8 6 0, show that e ±. marks To model the shape of the shade sail, the domain is restricted to R+ {0}. The rectangular hperola that follows this constraint is shown on the set of aes at right. 0 On the graph aove right, clearl lael the foci point and the asmptotes. marks To complete the shape of the shade sail, two line segments ( and are drawn as shown on the graph at right. The line segments egin at (0, 0 and end at the point of intersections with the hperola. Line is found reflecting in the -ais. c Eplain wh the gradient of must lie etween 0 and. marks 0 d If the length of is 5, show that the point of intersection of and the rectangular hperola is (,. marks e If the length of is also 5, determine the equation of. marks The shade sails are made Cover All Shades. There are two tpes of material that shade sails can e made from. Each tpe of material is descried the amount of ultraviolet (UV ras that can penetrate through the fires. The two materials are defined as light protection (l and maimum protection (m. The cost per metre of the light protection faric is $.75 and the cost per metre of the maimum protection faric is $.50. Cover All Shades keeps 00 metres of light protection material and 00 metres of maimum protection material in stock. Each sail requires 5 metres of material to construct. A maimum of sails are made each week. At least light protection sails are made each week. The information aove can e represented the following inequalities: Inequalit : l 0 Inequalit : m 0 Inequalit : l + m 0 Inequalit : l 60 a An additional Inequalit 5 is descried as m l. In the contet of this prolem, eplain Inequalit 5. marks Lines to 5 are shown on the graph at right. Clearl show the feasile region. mark c Write down the coordinates of all oundar points for the feasile m 0 region found in part. marks d Determine the ojective function, S, for this prolem. Write our answer as an equation in terms of S, l and m. marks e Determine the maimum weekl sales, in dollars, that Cover All 0 Shades can epect. Write our answer to the nearest cent. marks f Outdoors Living places an order for 6 light protection and 9 maimum protection sails. Eplain, in the contet of this prolem, if Cover All Shades is ale to deliver this order within a week l marks Solutions Eam practice eam practice 6