7Coordinate. geometry UNCORRECTED PAGE PROOFS. 7.1 Kick off with CAS

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1 7.1 Kick off with CAS 7Coordinate geometry 7. Distance between two points 7.3 Midpoint of a line segment 7.4 Parallel lines and perpendicular lines 7.5 Applications 7.6 Review

2 7.1 Kick off with CAS U N C O R R EC TE D PA G E PR O O FS To come Please refer to the Resources tab in the Prelims section of your ebookplus for a comprehensive step-by-step guide on how to use your CAS technology. c07coordinategeometry.indd 53 6/1/15 8:0 PM

3 7. WorKEd EXAMPLE 1 Distance between two points Coordinate geometry is a branch of mathematics with many practical applications. The distance between two points can be calculated easily using Pythagoras theorem. It is particularly useful when trying to find a distance that is difficult to measure directly; for example, finding the distance from a point on one side of a lake to a point on the other side. Let A (x 1, y 1 ) and B (x, y ) be two points on the Cartesian plane as shown below. Triangle ABC is a right-angled triangle. AC = x x 1 BC = y y 1 By Pythagoras theorem: AB = AC + BC = (x x 1 ) + (y y 1 ) Hence AB = "(x x 1 ) + (y y 1 ) The distance between two points A (x 1, y 1 ) and B (x, y ) is: AB = "(x x 1 ) + (y y 1 ) a Find the distance between the points A and B in the figure at right. b Find the distance between the points P ( 1, 5) and Q (3, ). WritE a 1 From the graph find points A and B. A ( 3, 1) and B (3, 4) Let A have coordinates (x 1, y 1 ). Let (x 1, y 1 ) = ( 3, 1). 3 Let B have coordinates (x, y ). Let (x, y ) = (3, 4). 4 Find the length AB by applying the formula for the distance between two points. A y y y 1 A (x 1, y 1 ) AB = "(x x 1 ) + (y y 1 ) = "[3 ( 3)] + (4 1) = "(6) + (3) =! =!45 = 3!5 = correct to decimal places x 1 B(x, y ) A 3 4 y 1 x C x 3 x B 54 MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Units 1 and

4 b 1 Let P have coordinates (x 1, y 1 ). Let (x 1, y 1 ) = ( 1, 5). Let Q have coordinates (x, y ). Let (x, y ) = (3, ). 3 Find the length PQ by applying the formula for the distance between two points. WorKEd EXAMPLE Tutorial eles 150 Worked example PQ = "(x x 1 ) + (y y 1 ) = "[3 ( 1)] + ( 5) = "(4) + ( 7) =! =!65 = correct to decimal places Prove that the points A (1, 1), B (3, 1) and C ( 1, 3) are the vertices of an isosceles triangle. 1 Plot the points. Note: For triangle ABC to be isosceles, two sides must have the same magnitude. Find the length AC. WritE/drAW y A x 1 1 C 3 From the diagram, AC appears to have the same length as BC. AC = "[1 ( 1)] + [1 ( 3)] = "() + (4) =!0 =!5 3 Find the length BC. BC = "[3 ( 1)] + [ 1 ( 3)] = "(4) + () =!0 =!5 4 Find the length AB. AB = "[3 (1)] + [ 1 (1)] = "() + ( ) =!4 + 4 =!8 =! 5 State your proof. Since AC = BC, triangle ABC is an isosceles triangle. Topic 7 CoordInATE geometry 55

5 Exercise 7. PRactise Consolidate Digital doc doc-998 Spreadsheet Distance between two points Distance between two points 1 WE1 a Find the distance between the points A and B shown at right. b Find the distance between the points. (, 5), (6, 8). a Find the distance between the points C and D shown at right. b Find the distance between the points ( 1, ) and (4, 14). 3 WE Prove that the points A (0, 3), B (, 1) and C (4, 3) are the vertices of an isosceles triangle. 4 Prove that the points A (3, 1), B ( 3, 7) and C ( 1, 3) are the vertices of an isosceles triangle. 5 The points P (, 1), Q ( 4, 1) and R ( 1, 3!3 1) are joined to form a triangle. Prove that triangle PQR is equilateral. 6 Prove that the quadrilateral with vertices A ( 1, 3), B (5, 3), C (1, 0) and D ( 5, 0) is a parallelogram. 7 Prove that the triangle with vertices D (5, 6), E (9, 3) and F (5, 3) is a rightangled triangle. 8 The vertices of a quadrilateral are A (1, 4), B ( 1, 8), C (1, 9) and D (3, 5). a Find the lengths of the sides. b Find the lengths of the diagonals. c What type of quadrilateral is it? 9 Calculate the distance between each of the pairs of points below, accurate to 3 decimal places. a ( 14, 10) and ( 8, 14) b (6, 7) and (13, 6) c ( 11, 1) and (, ) 10 Find the distance between each of the following pairs of points in terms of the given variables. a (a, 1), (, 3) b (5, 6), (0, b) c (c, ), (4, c) d (d, d), (1, 5) 11 If the distance between the points (3, b) and ( 5, ) is 10 units, then the value of b is: A 8 b 4 c 4 d 0 e 1 A rhombus has vertices A (1, 6), B (6, 6), C (, ) and D (x, y). The coordinates of D are: A (, 3) B (, 3) C (, 3) D (3, ) E (3, ) 13 A rectangle has vertices A (1, 5), B (10.6, z), C (7.6, 6.) and D (, 1). Find: a the length of CD b the length of AD c the length of the diagonal AC d the value of z. 14 Show that the triangle ABC with coordinates A (a, a), B (m, a) and C ( a, m) is isosceles. y A C B x D MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Units 1 and

6 Master 15 Two hikers are about to hike from A to B (shown on the map below). How far is it from A to B as the crow flies, that is, in a straight line? W 50 m 100 m 00 m N 100 m 00 m 300 m B (E7, N4) Lake Phillios E 7.3 A (W1, S5) Grid spacing : 1 km 16 Using the coordinates shown on the aerial photo of the golf course, calculate (to the nearest metre): a the horizontal distance travelled by the golf ball for the shot down the fairway b the horizontal distance that needs to be covered in the next shot to reach the point labelled A in the bunker. Midpoint of a line segment We can determine the coordinates of the midpoint of a line segment by applying the midpoint formula shown below. Midpoint formula Consider the line segment connecting the points A (x 1, y 1 ) and B (x, y ). Let P (x, y) be 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 x x 1 = x x x = x 1 + x x = x 1 + x Similarly it can be shown that y = y 1 + y. S y (in metres) (80, 64) A (30, 148) (5, 96) x (in metres) y B (x, y ) (y y) P (x, y) (x x) D (y y 1 ) A (x 1, y 1 )(x x C 1 ) x Topic 7 CoordinATE geometry 57

7 WorKEd EXAMPLE 3 In general, the coordinates of the midpoint of a line segment joining the points (x 1, y 1 ) and (x, y ) can be found by averaging the x- and y-coordinates of the end points, respectively. The coordinates of the midpoint of the line segment joining (x 1, y 1 ) and (x, y ) are: a x 1 + x, y 1 + y b. y (x 1, y 1 ) M (x, y ) x 1 + x ( y, 1 + y ) Find the coordinates of the midpoint of the line segment joining (, 5) and (7, 1). WritE 1 Label the given points (x 1, y 1 ) and (x, y ). Let (x 1, y 1 ) = (, 5) and (x, y ) = (7, 1). Find the x-coordinate of the midpoint. x = x 1 + x = + 7 = 5 = 1 3 Find the y-coordinate of the midpoint. y = y 1 + y = = 6 = 3 4 Give the coordinates of the midpoint. Hence, the coordinates of the midpoint are ( 1, 3). WorKEd EXAMPLE 4 The coordinates of the midpoint, M, of the line segment AB are (7, ). If the coordinates of A are (1, 4), find the coordinates of B. 1 Label the start of the line segment (x 1, y 1 ) and the midpoint (x, y). Find the x-coordinate of the end point. x = x 1 + x WritE/drAW Let (x 1, y 1 ) = (1, 4) and (x, y) = (7, ). 7 = 1 + x 14 = 1 + x x = 13 x 58 MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Units 1 and

8 3 Find the y-coordinate of the end point. y = y 1 + y Exercise 7.3 Digital doc doc-999 Spreadsheet Midpoint of a segment PRactise Consolidate Midpoint of a line segment 1 WE3 Find the coordinates of the midpoint of the line segment joining ( 5, 1) and ( 1, 8). = 4 + y 4 = 4 + y y = 8 4 Give the coordinates of the end point. Hence, the coordinates of the point B are (13, 8). 5 Check that the coordinates are feasible. y B 8 (13, 8) 4 M (7, ) x (1, 4) A Find the coordinates of the midpoint of the line segment joining (4, ), (11, ). 3 WE4 The coordinates of the midpoint M of the line segment AB are (, 3). If the coordinates of A are (7, 4), find the coordinates of B. 4 The coordinates of the midpoint M of the line segment AB are (, 4). If the coordinates of A are (1, 8), find the coordinates of B. 5 The vertices of a square are A (0, 0), B (, 4), C (6, ) and D (4, ). Find: a the coordinates of the centre of the square b the side length c the length of the diagonals. 6 The midpoint of the line segment joining the points (, 1) and (8, 3) is: A (6, ) B (5, ) C (6, ) D (3, 1) E (5, ) 7 If the midpoint of AB is ( 1, 5) and the coordinates of B are (3, 8), then A has coordinates: A (1, 6.5) B (, 13) C ( 5, ) D (4, 3) E (7, 11) 8 Find the coordinates of the midpoint of each of the following pairs of points, in terms of a variable or variables where appropriate. a (a, a), (6a, 5a) b (5, 3c), (11, 3c) c (3f, 5), (g, 1) 9 Find the value of a below so M is the midpoint of the segment joining points A and B. a A (, a), B ( 6, 5), M ( 4, 5) b A (a, 0), B (7, 3), M (8, 3 ) 10 a The vertices of a triangle are A (, 5), B (1, 3) and C ( 4, 3). 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. b Hence show that BC = PQ. Topic 7 CoordinATE geometry 59

9 Master a A quadrilateral has vertices A (6, ), B (4, 3), C ( 4, 3) and D (, ). Find: i the midpoint of the diagonal AC ii the midpoint of the diagonal BD. b Comment on your finding. 1 a The points A ( 5, 3.5), B (1, 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. b Describe the triangle ABC. What could PC represent? 13 Find the equation of the straight line that passes through the midpoint of A (, 5) and B (, 3) and has a gradient of Find the equation of the straight line that passes through the midpoint of A ( 1, 3) and B (3, 5) and has a gradient of A fun-run course is y drawn (not to scale) (1.5, 3.5) at right. If drink D1 stations D1, D and ( 4.5, 5) D3 are to be placed D Official tent at the middle of each x straight section, give (1.5, ) the map coordinates START/ FINISH of each drink station. 16 Find the equation of a line that has a gradient of 5 and passes Coordinates D3 (13, 8) are in kilometres. (3, 7) through the midpoint of the segment joining ( 1, 7) and (3, 3). Parallel lines and perpendicular lines Parallel Lines The equation of a straight line may be expressed in the form: y = mx + c where m is the gradient of the line and c is the y-intercept. The gradient can be calculated if two points, (x 1, y 1 ) and (x, y ), are given. m = y y 1 x x 1 An alternative form for the equation of a straight line is: ax + by + c = 0 where a, b and c are constants. Another alternative form is: y y 1 = m(x x 1 ) where m is the gradient and (x 1, y 1 ) is a point on the line. 60 MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Units 1 and

10 WorKEd EXAMPLE 5 Show that AB is parallel to CD given that A has coordinates ( 1, 5), B has coordinates (5, 7), C has coordinates ( 3, 1), and D has coordinates (4, 15). WritE 1 Find the gradient of AB. Let A ( 1, 5) = (x 1, y 1 ) and B (5, 7) = (x, y ). m = y y 1 x x 1 m AB = 7 ( 5) 5 ( 1) = 1 6 Find the gradient of CD. Let C ( 3, 1) = (x 1, y 1 ) and D (4, 15) = (x, y ). 3 Compare the gradients to determine if they are parallel. (Note: means is parallel to.) WorKEd EXAMPLE 6 Tutorial eles 153 Worked example 6 = m CD = 15 1 = ( 3) = Parallel lines have the same gradient. m AB = m CD =, hence AB CD. Collinear points lie on the same straight line. Show that the points A (, 0), B (4, 1) and C (10, 4) are collinear. WritE 1 Find the gradient of AB. Let A (, 0) = (x 1, y 1 ) and B (4, 1) = (x, y ). Since m = y y 1 x x 1 m AB = 1 0 = 1 4 Find the gradient of BC. Let B (4, 1) = (x 1, y 1 ) and C (10, 4) = (x, y ). m BC = = 3 6 = 1 Topic 7 CoordInATE geometry 61

11 3 Show that A, B and C are collinear. Since m AB = m BC = 1 then AB BC. Since B is common to both line segments, A, B and C must lie on the same straight line. That is, A, B and C are collinear. WorKEd EXAMPLE 7 Find the equation of the straight line that passes through the point (, 5) and is parallel to the line y = 3x 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 because the line is parallel to y = 3x + 1, the gradients will be the same. Use the formula y y 1 = m(x x 1 ) and substitute the coordinates of the point and the gradient to find the equation of the line. Perpendicular lines In this section, we examine some of the properties of perpendicular lines. Observing the graphs can be very useful in investigating these properties. Consider the diagram below, where the line segment AB is perpendicular to the line segment BC. Line AC is parallel to the x-axis. Line BD is the height of the resulting triangle ABC. y Let B m AB = m 1 = a b = tan (θ) Let m BC = m a = b = tan (θ) b = a 1 = Hence m 1 1 m = m 1 WritE Point on the line: (, 5) Gradient: m = 3. y y 1 = m(x + x 1 ) y 5 = 3(x ) y 5 = 3x 6 y = 3x 1 or m 1 m = 1 Hence, if two lines are perpendicular to each other, then the product of their gradients is 1. A θ b α a θ D c α x C 6 MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Units 1 and

12 Two lines are perpendicular if and only if: m 1 m = 1 or m = 1 m 1 WorKEd EXAMPLE 8 Tutorial eles 154 Worked example 8 ExErcisE 7.4 DIGITAL DOC doc-9933 Spreadsheet Gradient DIGITAL DOC doc-9935 Spreadsheet Perpendicular checker PrActisE Show that the lines y = 5x + and 5y x + 15 = 0 are perpendicular to one another. WritE 1 Find the gradient of equation 1. y = 5x + Hence m 1 = 5 Find the gradient of equation. 5y x + 15 = 0 Rewrite in the form y = mx + c: 5y = x 15 3 Test for perpendicularity. (The two lines are perpendicular if the product of their gradients is 1.) y = x 5 3 Hence m = 1 5 m 1 m = = 1 Hence, the two lines are perpendicular to each other. Parallel lines and perpendicular lines 1 WE5 Show that AB is parallel to CD given that A has coordinates (, 4), B has coordinates (8, 1), C has coordinates ( 6, ), and D has coordinates (, 6). Show that AB is parallel to CD given that A has coordinates (1, 0), B has coordinates (, 5), C has coordinates (3, 15), and D has coordinates (7, 35). 3 WE6 Show that the points A (0, ), B (5, 1) and C ( 5, 5) are collinear. 4 Show that the points A(3, 1), B(5, ) and C(11, 5) are collinear. 5 WE7 Find the equation of a straight line given the following conditions. The line passes through the point ( 1, 3) and is parallel to y = x Find the equation of a straight line given that the line passes through the point (4, 3) and is parallel to 3y + x = 3. 7 WE8 Show that the lines y = 6x 3 and x + 6y 6 = 0 are perpendicular to one another. Topic 7 CoordInATE geometry 63

13 Consolidate 8 Show that the lines y = x 4 and x + y 10 = 0 are perpendicular to one another. 9 Which pairs of the following straight lines are parallel? a x + y + 1 = 0 b y = 3x 1 c y x = 3 d y = 4x + 3 e y = x 1 f 6x y = 0 DIGITAL DOC doc-9935 Spreadsheet Parallel checker Master g 3y = x + 4 h y = 5 x 10 Show that the line that passes through the points ( 4, 9) and (0, 3) also passes through the point (6, 6). 11 In each of the following, show that ABCD is a parallelogram. a A (, 0), B (4, 3), C (, 4), D (0, 1) b A (, ), B (0, ), C (, 3), D (0, 1) c A (.5, 3.5), B (10, 4), C (.5,.5), D ( 5, 5) 1 In each of the following, show that ABCD is a trapezium. a A (0, 6), B (, ), C (0, 4), D ( 5, 9) b A (6, 3), B (18, 16), C (1, 1), D ( 3, 3) c A (, 7), B (1, 1), C ( 0.6,.6), D (, 3) 13 The line that passes through the points (0, 6) and (7, 8) also passes through: A (4, 3) B (5, 4) C (, 10) D (1, 8) E (1, 4) 14 The point ( 1, 5) lies on a line parallel to 4x + y + 5 = 0. Another point on the same line as ( 1, 5) is: A (, 9) B (4, ) C (4, 0) D (, 3) E (3, 11) 15 Determine which pairs of the following straight lines are perpendicular. a x + 3y 5 = 0 b y = 4x 7 c y = x d y = x + 1 e y = 3x + f x + 4y 9 = 0 g x + y = 6 h x + y = 0 16 Show that the following sets of points form the vertices of a rightangled triangle. a A (1, 4), B (, 3), C (4, 7) b A (3, 13), B (1, 3), C ( 4, 4) c A (0, 5), B (9, 1), C (3, 14) 17 Prove that the quadrilateral ABCD is a rectangle when A is (, 5), B (6, 1), C (3, ) and D ( 1, ). 18 Find the equation of the straight line that cuts the x-axis at 3 and is perpendicular to the line with equation 3y 6x = Calculate the value of m for which the following pairs of equations are (i) parallel (ii) perpendicular. a y 5x = 7 and 4y + 1 = mx b 5x 6y = 7 and 15 + mx = 3y 0 Prove that the quadrilateral ABCD is a rhombus, given A(, 3), B(3, 5), C(5, 6) and D(4, 4). Hint: The diagonals of a rhombus intersect at right angles. 64 MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Units 1 and

14 7.5 Applications In this section we look at two important applications: the equation of a straight line, and equations of horizontal and vertical lines. WorKEd EXAMPLE 9 The equation of a straight line The equation of a straight line can be determined by two methods. The y = mx + c method requires the gradient, m, and a given point to be known, in order to establish the value of c. Note: Because the value of c represents the y-intercept, it can be substituted directly if known. Find the equation of the straight line that passes through the point (3, 1) and is parallel to the straight line with equation y = x + 1. WritE 1 Write the general equation. y = mx + c Find the gradient of the given line. y = x + 1 has a gradient of Hence m =. 3 Substitute for m in the general equation. so y = x + c 4 Substitute the given point to find c. (x, y) = (3, 1) 1 = (3) + c = 6 + c c = 7 5 Substitute for c in the general equation. y = x 7 or x y 7 = 0 The alternative method comes from the gradient definition: m = y y 1 x x 1 m(x x 1 ) = y y 1 Hence Using the general point (x, y) instead of the specific point (x, y ) gives the general equation: y y 1 = m(x = x 1 ) This requires the gradient, m, and a given point (x 1, y 1 ) to be known. WorKEd EXAMPLE 10 Find the equation of the line that passes through the point (0, 3) and is perpendicular to a straight line with a gradient of 5. WritE 1 Find the gradient of the perpendicular line. Given m = 5 m 1 = 1 5 Topic 7 CoordInATE geometry 65

15 Substitute for m and (x 1, y 1 ) in the general equation. Since y y 1 = m(x x 1 ) and (x 1, y 1 ) = (0, 3) 1 then y 3 = (x 0) 5 x = 5 5(y 3) = x 5y 15 = x x + 5y 15 = 0 WorKEd EXAMPLE 11 Horizontal and vertical lines For horizontal lines the gradient is equal to zero, so the equation y = mx + c becomes y = c. Notice that x does not appear in the equation because there is no x-intercept. Horizontal lines are parallel to the x-axis. In the case of vertical lines, the gradient is infinite or undefined. The general equation for a vertical line is given by x = a. In this case, just as the equation suggests, a represents the x-intercept. Notice that y does not appear in the equation because there is no y-intercept. Vertical lines are parallel to the y-axis. The graphs of y = 4 and x = 3 are shown at right to highlight this information. Find the equation of: a the vertical line that passes through the point (, 3) b the horizontal line that passes through the point (, 6). a For a vertical line, there is no y-intercept, so y does not appear in the equation. The x-coordinate of the point is. b For a horizontal line, there is no x-intercept, so x does not appear in the equation. The y-coordinate of the point is 6. WorKEd EXAMPLE 1 WritE a x = b y = 6 y 4 0 x = 3 Find the equation of the perpendicular bisector of the line joining the points (0, 4) and (6, 5). 1 Find the gradient of the line joining the given points using the general equation. WritE Let (0, 4) = (x 1, y 1 ). Let (6, 5) = (x, y ). m = y y 1 x x 1 m = 5 ( 4) 6 0 = y 0 y = 4 x x = 3 66 MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Units 1 and

16 1 Find the gradient of the perpendicular line. For lines to be perpendicular, m =. m 1 3 Find the midpoint of the line joining the given points. 4 Substitute for m and (x 1, y 1 ) in the general equation. 5 Simplify by removing the fractions. (a) Multiply both sides by 3. (b) Multiply both sides by. Note: The diagram at right shows the geometric situation. WorKEd EXAMPLE 13 Tutorial eles 155 Worked example 13 m 1 = 3 x = x 1 + x = = 3 y = y 1 + y = = 1 Hence the coordinates of the midpoint are (3, 1 ). Since y y 1 = m(x x 1 ) and (x 1, y 1 ) = (3, 1 ) and m 1 = then y 1 = (x 3) 3(y 1 ) = (x 3) 3y 3 = x + 6 6y 3 = 4x + 1 4x + 6y 15 = 0 y (6, 5) x 4 ABCD is a parallelogram. The coordinates of A, B and C are (1, 5), (4, ) and (, ) respectively. Find: a the equation of AD c the coordinates of D. 3 3 b the equation of DC a 1 Draw the parallelogram ABCD. Note: The order of the lettering of the geometric shape determines the links in the diagram. For example: ABCD means join A to B to C to D to A. This avoids any ambiguity. WritE/drAW a y A 5 D C B Topic 7 CoordInATE geometry 67

17 Find the gradient of BC. m BC = 4 = 4 = 3 State the gradient of AD. Since m BC = and AD BC then m AD =. 4 Using the given coordinates of A and the gradient of AD, find the equation of AD. Exercise 7.5 Digital doc doc-9936 Spreadsheet Equation of a straight line PRactise Applications y = x + c Let (x, y) = (1, 5): 5 = (1) + c c = 3 Hence, the equation of AD is y = x + 3. b 1 Find the gradient of AB. b m AB = = 3 3 = 1 State the gradient of DC. Since m AB = 1 and DC AB then m DC = 1. 3 Using the given coordinates of C and the gradient of DC, find the equation of DC. c To find D, solve simultaneously the point of intersection of the equations AD and DC. Note: Alternatively, a calculator could be used to determine the point of intersection of AD. y = x + c Let (x, y) = (, ): = () + c c = 0 Hence, the equation of DC is y = x. c Equation of AD: y = x + 3 [1] Equation of DC: y = x [] [1] []: 0 = 3x + 3 3x = 3 x = 1 Substituting x = 1 in []: y = ( 1) = 1 Hence, the coordinates of D are ( 1, 1). 1 WE9 Find the equation of the straight line that passes through the point (4, 1) and is parallel to the straight line with equation y = x 5. Find the equation of the line that passes through the point (3, 4) and is parallel to the straight line with equation y = x 5. 3 WE10 Find the equation of the line that passes through the point (, 7) and is perpendicular to a straight line with a gradient of. 3 4 Find the equation of the line that passes through the point (, 0) and is perpendicular to a straight line with a gradient of. 5 WE11 Find the equation of: a the vertical line that passes through the point (1, 8) b the horizontal line that passes through the point ( 5, 7). 68 MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Units 1 and

18 Consolidate 6 Find the equation of: a the vertical line that passes through the point ( 1, 4) b the horizontal line that passes through the point (5, ). 7 WE1 Find the equation of the perpendicular bisector of the line joining the points (1, ) and ( 5, 4). 8 Find the equation of the perpendicular bisector of the line joining the points ( 4, 0) and (, 6). 9 WE13 ABCD is a parallelogram. The coordinates of A, B and C are (4, 1), (1, ) and (, 1) respectively. Find: a the equation of AD b the equation of DC c the coordinates of D. 10 ABCD is a parallelogram. The coordinates of A, B and C are Q 1, 5 R, (1, 1) and 3 3 Q 3, 1R respectively. Find: a the equation of AD b the equation of DC c the coordinates of D. 11 Find the equations of the following straight lines. a Gradient 3 and passing through the point (1, 5) b Gradient 4 and passing through the point (, 1) c Passing through the points (, 1) and (4, ) d Passing through the points (1, 3) and (6, 5) e Passing through the point (5, ) and parallel to x + 5y + 5 = 0 f Passing through the point (1, 6) and parallel to x 3y = 0 g Passing through the point ( 1, 5) and perpendicular to 3x + y + = 0 1 Find the equation of the line that passes through the point (, 1) and is: a parallel to the straight line with equation x y 3 = 0 b perpendicular to the straight line with equation x y 3 = Find the equation of the line that contains the point (1, 1) and is: a parallel to the straight line with equation 3x 5y = 0 b perpendicular to the straight line with equation 3x 5y = a The vertical line passing through the point (3, 4) is given by: A y = 4 B x = 3 C y = 3x 4 D y = 4x + 3 E x = 4 15 Which of the following points does the horizontal line given by the equation y = 5 pass through? A ( 5, 4) B (4, 5) C (3, 5) D (5, 4) E (5, 5) 16 Which of the following statements is true? A Vertical lines have a gradient of zero. B The y-coordinates of all points on a vertical line are the same. C Horizontal lines have an undefined gradient. D The x-coordinates of all points on a vertical line are the same. E A horizontal line has the general equation x = a. 17 Which of the following statements is false? A Horizontal lines have a gradient of zero. B The straight line joining the points (1, 1) and ( 7, 1) is vertical. C Vertical lines have an undefined gradient. D The straight line joining the points (1, 1) and ( 7, 1) is horizontal. E A horizontal line has the general equation y = c. Topic 7 CoordinATE geometry 69

19 Master 18 The triangle ABC has vertices A (9, ), B (3, 6) and C (1, 4). a Find the midpoint, M, of BC. b Find the gradient of BC. c Show that AM is the perpendicular bisector of BC. d Describe triangle ABC. 19 Find the equation of the perpendicular bisector of the line joining the points (, 9) and (4, 0). 0 a The equation of the line passing through the point (4, 3) and parallel to the line x 4y + 1 = 0 is: A x y + = 0 B x y 5 = 0 C x y 10 = 0 D x y 11 = 0 E y + x + = 0 b The equation of the perpendicular bisector of the line segment AB where A is ( 3, 5) and B is (1, 7) is: A y = x + 13 B y = x 8 C y = x + 11 D y = x + 4 E y = x 4 c The coordinates of the centroid of triangle ABC with vertices A (1, 8), B (9, 6) and C ( 1, 4) are: A (4, 5) B (0, 6) C (3, 6) D (5, 7) E (, 7) 1 The map at right shows the proposed course for a yacht race. Buoys have been positioned at A (1, 5), B (8, 8) and C (1, 6), but the last buoy's placement, D (10, w), is yet to be finalised. a How far is the first stage of the race, that is, from the start, O, to buoy A? b The race marshall boat, M, is situated halfway between buoys A and C. What are the coordinates of the boat? y 11 c Stage 4 of the race (from C to D) is 10 Scale: 1 unit 1 km N perpendicular to stage 3 (from B to 9 Buoy B C). What is the gradient of CD? 8 d Find the linear equation that 7 Buoy describes stage 4. 6 A Buoy C M e Hence determine the exact position 5 4 of buoy D. 3 E f An emergency boat is to be placed at point E, H Buoy D of the way from buoy 1 3 O A to buoy D. Into what internal ratio (Start) x does point E divide the distance from A to D? g Determine the coordinates of the emergency boat. h How far is the emergency boat from the hospital, located at H, km north of the start? To supply cities with water when the source is a long distance away, artificial channels called aqueducts may be built. More than 000 years after it was built, a Roman aqueduct still stands in southern France. It brought water from a source in Uzès to the city of Nîmes. The aqueduct does not follow a direct route between these two locations as there is a mountain range between them. The table shows the approximate distance from Uzès along the aqueduct to each town (or in the case of Pont du Gard, a bridge) and the aqueduct's height above sea level at each location. 70 MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Units 1 and

20 Location Distance from Uzès (km) Height of aqueduct above sea level (m) Uzès 0 76 Pont du Gard (bridge) St. Bonnet 5 64 St. Gervasy Nîmes a Show the information in the table as a graph with the distance from Uzès along the horizontal axis. Join the plotted points with straight lines. b Calculate the gradient of the steepest part of the aqueduct (in m/km). c Suppose the aqueduct started at Uzès and ended at Nîmes but had a constant gradient. Write a linear equation to describe its course. d Using the equation found in part c, calculate the height of the aqueduct at the Pont du Gard. This calculated height is higher than the actual height. How much higher? e Why do you think the Romans made the first part of the aqueduct steeper than the rest? Topic 7 CoordInATE geometry 71

21 ONLINE ONLY 7.6 Review the Maths Quest review is available in a customisable format for you to demonstrate your knowledge of this topic. the review contains: short-answer questions providing you with the opportunity to demonstrate the skills you have developed to efficiently answer questions using the most appropriate methods Multiple-choice questions providing you with the opportunity to practise answering questions using CAS technology studyon is an interactive and highly visual online tool that helps you to clearly identify strengths and weaknesses prior to your exams. You can then confidently target areas of greatest need, enabling you to achieve your best results. Extended-response questions providing you with the opportunity to practise exam-style questions. A summary of the key points covered in this topic is also available as a digital document. REVIEW QUESTIONS Download the Review questions document from the links found in the Resources section of your ebookplus. Units 1 & AOS # Topic 7 Concept # Coordinate geometry Sit Topic test 7 MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Units 1 and

22 7 Answers Exercise 7. 1 a AB = 5 b 5 a CD =!10 or 6.3 b 13 3, 4, 5, 6 and 7 Answers will vary. 8 a AB = 4.47, BC =.4, CD = 4.47, DA =.4 b AC = 5, BD = 5 c Rectangle 9 a 7.11 b c a "a 4a B 1 D b "b 1b + 61 c "1c 6c + 10 d "5d d a 1 b 5 c 13 d. 14 Answers will vary km 16 a 16 b 108 Exercise ( 3, 3 1 ) (7 1, 0) 3 ( 3, 10) 4 ( 5, 0) 5 a (3, 1) 6 D 7 C b 4.47 c a (4a, 3a) b (8, 3c) c a 3f + g, b 9 a 5 b 9 10 a i ( 1, 4) ii (1 1, 1) iii 3.9 iv 7.8 b Answers will vary. 11 a i (1, 0.5) ii (1, 0.5) b Answers will vary. 1 a i (, ) ii 8.94 iii 9.55 iv 9.55 b Isosceles triangle, height 13 y = 3x 14 3y x + 14 = 0 15 D1 ( 1.5, 4.5), D ( 1.5, 1.5), D3 (8, 7.5) 16 y = 5x 7 Exercise 7.4 1,, 3 and 4 Answers will vary. 5 5y = x y + x + 1 = 0 7 and 8 Answers will vary. 9 b, f; c, e 10, 11, 1 Answers will vary. 13 B 14 E 15 a, e; b, f; c, h; d, g 16, 17 Answers will vary y = x a m = b m = 18 0 Answers will vary. 5 5 Exercise y = x 9 y = x 1 3 3x + y 8 = 0 4 x y = 0 5 a x = 1 b y = 7 6 a x = 1 b y = Topic 7 CoordinATE geometry 73

23 7 y = x 3 8 y = x 9 a y = x + 5 b y = x + 3 c (1, 4) 10 a y = 4x 3 b y = x 4 c 1 1, a y = 3x + b y = 4x + 9 c 3x y 8 = 0 d x + 5y + 13 = 0 e x + 5y + 5 = 0 f x 3y + 17 = 0 g x 3y 14 = 0 1 a x y + 5 = 0 b x + y = 0 13 a 3x 5y + = 0 14 B 15 C 16 D 17 B b 5x + 3y 8 = 0 18 a (, 5) b 1 c Answers will vary. d Isosceles triangle 19 4x 6y + 3 = 0 0 a A b D c C 1 a 5.10 km a b (6.5, 5.5) c d y = x 18 e (10, ) f :1 g (7, 3) h km Height of aqueduct above sea level (m) b 0.69 m/km c Distance from Uzès (km) y = 0.34x + 76, where 76 is the height in metres above sea level and x is the distance in km from Uzès d 5.56 m e Check with the teacher. 74 MATHS QUEST 11 SPECIALIST MATHEMATICS VCE UNITS 1 AND

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