0 COORDINATE GEOMETRY

Transcription

1 0 COORDINATE GEOMETRY Coordinate Geometr 0-1 Equations of Lines 0- Parallel and Perpendicular Lines 0- Intersecting Lines 0- Midpoints, Distance Formula, Segment Lengths 0- Equations of Circles 0-6 Problem Solving Table of Contents / 1

2 0-1 Equations of Lines 0-1 Equations of Lines / Slope can be considered to be a measure of the "steepness" of a line. Hopefull, ou recall that ou can find the slope of the line connecting two points b the following formula: 1 m Remember that linear equations can be written in a variet of was. Look at the graph shown below. 1 Here are just a few of the infinitel man equations for the line shown above: 0 ( ) There are a few forms of linear equations that are useful enough that we give them names. Slope-intercept form: m b (this is useful because m gives the slope, and b gives the -int) Point-slope form: 1 m( 1 ) (this is useful because m is the slope and ( 1, 1) is on the line) Standard-form: A B C Eample 1 Write the equation 6 in slope-intercept form. Then state its slope and -intercept and sketch a graph.

3 0-1 Equations of Lines /

4 Eample Write an equation for the line shown below using slope-intercept form. 0-1 Equations of Lines / Eample Write an equation for the line shown below using point-slope form. Eample Write the equation 1 in standard form with integer coefficients.

5 Eample Find the -intercept and -intercept of the graph of the equation 6. Then sketch a graph of the equation. 0-1 Equations of Lines / Written Eercises A In problems 1-10 sketch graphs of each equation ( ). 1 ( ) ( 1)

6 0-1 Equations of Lines / 6 In problems 11-1 write equations for each graph in the form indicated. 11. in slope-intercept form 1. in slope-intercept form 1. in point-slope form 1. in point-slope form 1. Find the slope of the line that passes through (-,) and (,6) Find the slope of the line that passes through, and,. 17. Find the slope of the line that passes through (1.,) and (,.) Find the slope of the line that passes through, and,1. B 19. Write an equation of the line that passes through (,) and (-,6) in slope-intercept form. 0. Write an equation of the line that passes through (,) and (-,) in slope-intercept form. 1. Write an equation of the line with -intercept of and slope - in slope-intercept form.. Write an equation of the line with -intercept of and slope in slope-intercept form.. Write an equation of the vertical line going through the point (,).. Write an equation of the horizontal line going through the point (,).. Write an equation of the line that passes through (, ) and (, 16). 6. Write an equation of the line that passes through (, ) and (-, ). 7. a) Write an equation for a line whose -intercept is / and whose -intercept is -7/9. b) Put the equation from part (a) in standard form with integer coefficients. 8. a) Write an equation for a line whose -intercept is / and which passes through (,-). b) Put the equation from part (a) in standard form with integer coefficients.

7 0-1 Equations of Lines / 7 9. Given the line 1 a) Find the -intercept and the -intercept. b) Sketch a graph of the equation. 0. Put the following equation in slope-intercept form: 1 1. ( 1) ( ) Put the following equation in slope-intercept form:. 1 Put the following equation in standard form with integer coefficients: 6. Put the following equation in standard form with integer coefficients: ( ) C. Suppose the graph of k d is a line with a negative intercept and a positive -intercept. a) Is it possible to determine the sign of k? If so, what is it? b) Is it possible to determine the sign of d? If so, what is it?. Suppose the graph of A B is a line with a negative slope and a positive -intercept. a) Is it possible to determine the sign of B? If so, what is it? b) Is it possible to determine the sign of AB? If so, what is it? 6. Given the equation of the line 1 a b a) Find the -intercept. b) Find the -intercept. c) Suppose the -intercept of a line is and the -intercept is. Using the equation above, quickl find an equation of the line. 0- Parallel and Perpendicular Lines Parallel lines are lines in the same plane that never intersect. Parallel lines have the same slope, ecept in the case of two vertical lines. Two vertical lines are parallel, but slope is undefined for vertical lines. In the left figure, the two lines are parallel; both have a slope of /. In the right figure both lines are vertical. The never intersect, so the are parallel, but neither has a defined slope. So, we can summarize: If two lines are parallel, then either the both have the same slope or both are vertical.

8 0- Parallel and Perpendicular Lines / 8 Perpendicular lines are lines that form right angles when the intersect. Perpendicular lines have slopes that are negative reciprocals of each other, ecept in the case of a vertical line and a horizontal line. A vertical line and a horizontal line are perpendicular, but the slope of a vertical line is undefined. In the above left figure, the two lines are perpendicular; one has a slope of / and the other a slope of -/. In the above right figure there are two lines, one horizontal and one vertical. The intersect at a right angle, so the are perpendicular, but the slope of the vertical line is undefined. So, we can summarize: If two lines are perpendicular, then either the have negative reciprocal slopes or one is horizontal and the other is vertical. Cool fact: You can tell that two numbers are negative reciprocals if their product is -1. Sometimes it is not necessar to know this fact, because it is often obvious that two numbers are negative reciprocals (e.g. / and -/). However, / and are also negative reciprocals even though the don t look as if the are. We can check the truth of this statement b simplifing the negative reciprocal of one to reveal the other or b multipling the two numbers together and observing that the product is -1. Eample 1 Line l has equation. n is parallel to l, and p is perpendicular to n. Find the slopes of n and p.

9 0- Parallel and Perpendicular Lines / 9 Eample Line l has -intercept (0,-17) and is parallel to the line whose equation is ( ). Find an equation for l. Eample Consider line l whose equation is 1. Find an equation of the line perpendicular to l so that the line passes through the point (,). Eample Find an equation of a line which passes through the origin and is perpendicular to.

10 Eample Show that two lines with slopes 1 and 1 are perpendicular. 0- Parallel and Perpendicular Lines / 10 Written Eercises A 1. If line l has slope /: a) What is the slope of a line parallel to l? b) What is the slope of a line perpendicular to l?. If line m has slope 1 / : a) What is the slope of a line parallel to m? b) What is the slope of a line perpendicular to m?. Consider the line whose equation is. a) Find the slope-intercept form of the equation of a perpendicular line with the same -intercept. b) Find a point-slope form of the equation of a parallel line which passes through (, -6).. Consider the line whose equation is 6. a) Find the slope-intercept form of the equation of a perpendicular line with the same -intercept. b) Find a point-slope form of the equation of a parallel line which passes through (, 176).. Consider the line whose equation is ( 8). a) Find the slope-intercept form of the equation of a perpendicular line with the same -intercept. b) Find a point-slope form of the equation of a parallel line which passes through (-, 1).

11 0- Parallel and Perpendicular Lines / For each slope, find the perpendicular slope and simplif full (rationalize denominator and reduce). a) b) c) d) a b 7. For each slope, find the perpendicular slope and simplif full (rationalize denominator and reduce). a) 1 b) c) 8. Decide whether the given lines are parallel, perpendicular, or neither. a) 1 b) c) d) B 9. Given the line whose equation is : a) Sketch a graph of the equation. b) Find the equation of a line parallel to the line from (a) whose -intercept is -. c) Find the equation of the line perpendicular to the line from (a) whose -intercept is -. d) Sketch graphs the lines from (b) and (c). 10. Given the line whose equation is 10 : a) Sketch a graph of the equation. b) Find the equation of a line parallel to the line from (a) whose -intercept is -. c) Find the equation of the line perpendicular to the line from (a) whose -intercept is -. d) Sketch graphs the lines from (b) and (c). 11. Find an equation in point-slope form for a line that contains point P and is parallel to l. a) P ( 0, ); l : b) P ( 1,); l : 1. Find an equation in point-slope form for a line that contains point P and is perpendicular to l. a) P (,0); l : b) P (, ); l : 1 1. Find an equation in standard form for a line that contains point P and is parallel to l. a) P (,); l : b) P (,); l : Find an equation in standard form for a line that contains point P and is perpendicular to l. a) P( 0, ); l : b) P (,); l : 1. Algebra Review. Simplif each radical b pulling out perfect squares. a) 0 b) 7 c) 80 d) Algebra Review. Simplif each radical b pulling out perfect squares. a) 00 b) 8 c) 1 8 d) d) c c

12 0- Parallel and Perpendicular Lines / Algebra Review. Rationalize the denominators of each fraction and simplif a) b) c) d) Algebra Review. Rationalize the denominators of each fraction and simplif. 1 a) b) c) d) Show that two lines with slopes 1 and are parallel. 0. Show that two lines with slopes ( 1 ) and ( 1 ) are perpendicular. 1. For each: if possible, provide an eample; if not possible eplain wh the situation is impossible. a) Two parallel lines, both with undefined slopes. b) Two perpendicular lines, both with undefined slopes. c) Two parallel lines, eactl one of which has an undefined slope. d) Two perpendicular lines, eactl one of which has an undefined slope. e) Two parallel lines with the same -intercept. f) Two perpendicular lines with the same -intercept. g) Two parallel lines with the same -intercept. h) Two perpendicular lines with the same -intercept. i) Two parallel lines, both of which pass through (1, -). j) Two perpendicular lines, both of which pass through (1, -). C. The slope of line l is. a) What is the slope (in simplest form) of a line perpendicular to l? b) Find the equation of a line which is perpendicular to l and passes through ( 1, ).. Prove the fact that if two numbers are negative reciprocals then their product is negative one and vice versa.

13 0- Intersecting Lines 0- Intersecting Lines / 1 Finding points of intersection between two graphs is as eas as solving the sstem of equations containing the equations of both graphs. This should make sense: We want to look for a point on Graph 1 (which is a solution to its equation) that is also a point on Graph (which is a solution to its equation). Thus, all points on both graphs (i.e. points of intersection) are solutions to both equations (i.e. solutions to the sstem containing both equations). There are a few methods for solving sstems of equations. Graphing Method. The graphing method involves just looking at graphs and inspecting for a point of intersection. A benefit of the graphing method is that it is eas to use if ou have the graphs. A drawback is that since ou are using our ees, the solution ou see is just an estimate. Another drawback is that sometimes ou don't alwas know where to look. You alwas should check our solution b plugging in (or at least realize that our solution is onl a good / approimate guess.) For the sstem graphed on the left, it looks like the lines intersect at (,) which is in fact the solution to the sstem so the graphing method would be useful in that case. However, in the sstem graphed on the right, the lines are prett close to parallel, and the point of intersection is far awa from our window of observation. In this situation the graphing method is relativel useless.

14 0- Intersecting Lines / 1 Substitution Method. The substitution method is nice whenever one equation can be used to plug-in nicel into the other to eliminate a variable. For eample, consider the sstem: Even though this is a non-linear(!) sstem, it is still eas to solve since when we plug in for in the second equation, we are left with a simple equation with just 's to solve: ( ). Once we solve for, we can plug back in to solve for. A benefit of this method is that it is eas to understand how it works. A drawback is that sometimes it is hard to get a variable isolated so that ou can plug it in somewhere. Linear Combinations Method. The linear combination method works b putting two linear equations in standard form and then adding appropriate multiples of the equation so that one variable cancels out. Here is an eample: 1 1 First, we just multipl the second equation b to eliminate fractions. 1 Now, we subtract from both sides of the second equations. 1 Now, we multipl the first equation b - so that the 's cancel once we add the equations. 8 8 The onl remaining step to get a simple equation to solve is to add the equations. 11 A benefit of this method is that it is eas to do in terms of arithmetic. A drawback is that it works onl (well, almost onl) with sstems in which all equations are linear.

15 Eample 1 Estimate the solution to 1 b graphing. Check our solution algebraicall Intersecting Lines / 1

16 0- Intersecting Lines / 16 Eample Find the point of intersection between the graphs of 1 and 6 b substitution. Eample Find the point of intersection between the graphs of and 1 b linear combinations.

17 0- Intersecting Lines / 17 Written Eercises A In problems 1- use the graph to help ou guess the solution to each sstem. Then confirm that our guesses are correct b plugging in to both equations ) ( In problems -8 solve the given sstem b substitution ) ( 1 In problems 9-1 solve the given sstem b linear combinations

18 0- Intersecting Lines / 18 B In problems 1- solve the given sstem b a method of our choosing ) (. 1) ( C In - solve the sstems using the graphing method and the graphs provided In problems -6 solve the given sstem b substitution Consider the line which passes through the origin and which is perpendicular to the line whose equation is 1. Where does this line intersect the horizontal line? 8. Consider the line which passes through the point (,) and which is parallel to the line whose equation is. Where does this line intersect the vertical line?

19 0- Midpoints, Distance Formula, Segment Lengths / Midpoints, Distance Formula, Segment Lengths A midpoint is the point halfwa between the endpoints of a line segment. For eample, (1,) is the midpoint of the segment with endpoints at (1,1) and (1,). The formula for midpoint is intuitivel eas to understand, because all ou have to do is take the average -coordinate and the average -coordinate: If 1 1 A ( 1, 1) and B(, ), then the midpoint of AB is,. The distance between two points is found b using the Pthagorean Theorem. Consider the points (,1) and (6,) as shown here: As ou can see, the distance between the two points is the length of the hpotenuse of the right triangle shown. Because the legs are horizontal and vertical, we can figure out their lengths simpl b subtracting: Their lengths are 6 = and 1 =. Now, using these lengths and the Pthagorean Theorem, we can find an epression for the length of the hpotenuse (note that the length of the hpotenuse is the distance between the points): (6 ) ( 1) 1. This process can be generalized to ield the distance formula: If A( 1, 1) and B(, ), then thedistance between Aand B is ( 1 ) ( 1 ). We define the length of a line segment to be the distance between its endpoints.

20 Eample 1 Find the midpoint and length of a segment whose endpoints are (-,) and (6,). 0- Midpoints, Distance Formula, Segment Lengths / 0 Eample A segment has one endpoint at (1,), and its midpoint is at (6,-). What is its other endpoint? Eample Find the equation of the perpendicular bisector (bisector means that it passes through the midpoint) of a segment whose endpoints are (1,) and (,-1).

21 0- Midpoints, Distance Formula, Segment Lengths / 1 Written Eercises A 1. If A(-,) and B(,), find the length and midpoint of AB?. If C(-,0) and D(,-), find the length and midpoint of CD?. What point is halfwa between (6,8) and (-,)?. What point is halfwa between (,-6) and (-6,)?. If M(,) is the midpoint of AB and A(,1), what are the coordinates of B? 6. If N(-,) is the midpoint of CD and C(0,6), what are the coordinates of D? 7. Suppose A(,) and B(,). a) What is the slope of the line containing AB? b) What is the midpoint of AB? c) Find an equation for the perpendicular bisector of AB. 8. Suppose A(-6,10) and B(,). a) What is the slope of the line containing AB? b) What is the midpoint of AB? c) Find an equation for the perpendicular bisector of AB. 9. Suppose A(-,) and B(,6). a) What is the slope of the line containing AB? b) What is the midpoint of AB? c) Find an equation for the perpendicular bisector of AB. B 10. Which is farther from the origin: (,) or (-,)? 11. Which is farther from (6,-): (,6) or (6,)? 1. Find at least five different points that are awa from (,). 1. The origin is 1 awa from (1,). Find at least five other points that are 1 awa from (1,). 1. Find an equation for the perpendicular bisector of DC assuming D(,) and C(8,). 1. Find an equation for the perpendicular bisector of CD assuming C(10,1) and D(6,). 16. Find an equation for the perpendicular bisector of DC assuming C(-,) and D(,). In 17-19, decide whether the triangle is scalene (no equal side lengths), isosceles (two equal side lengths), or equilateral (three equal side lengths). 17. Triangle ABC where A(,), B(6,), and C(,7). 18. Triangle DEF where D(,1), E(,-), and F(,). 19. Triangle MNP where M(0,0), N(6,0), and P(, 7 ).

22 0- Midpoints, Distance Formula, Segment Lengths / C 0. Two vertices of square ABCD are A(,) and B(,-1). Find possible coordinates for C and D. 1. Consider A(,) and B(7,9). a) Find the perpendicular bisector of AB. b) Pick an point on the perpendicular bisector (other than the midpoint of AB ) and call it P. c) Find the lengths of AP and BP. d) Eplain the result from (c).. The vertices of a triangle are A(0,), B(10,-), and C(,10). a) Find the length of AB. b) Find the length of the segment connecting the midpoints of AC and BC. c) Find the slope of the line contains AB. d) Find the slope of the line that passes through the midpoints of AC and BC.. Let A(0,0), B(7,1), C(1,6), and D(,). a) Plot the points on the same graph and connect them to form a quadrilateral ABCD. b) Verif that all four sides have the same length. c) Find the slopes of each segment. What do ou notice? d) Find the midpoints of AB, BC, CD, and DA. Plot the points on our graph and label them X, Y, Z, and W. e) What does XYZW look like? Check slopes to see if our answer is correct.

23 0- Equations of Circles 0- Equations of Circles / A circle is a collection of points that are a set distance awa from a point called its center. A radius is a line segment that connects the center to an point on the circle. Obviousl, the length of the radius is equal to the aforementioned "set distance". If r is the length of the radius, we can come up with an equation that describes a point (, ) on the circle if we assume that the center of the circle is (a, b). With the distance formula we can write a statement saing that the distance between (, ) (which is the point on the circle) to (a, b) (which is the center of the circle) is equal to r: ( a) ( b) r. If we square both sides, we are left with the equation of a circle in standard form: ( a) ( b) r So, if we see the equation ( ) ( ), we know that it is the equation of a circle whose center is (, ) and whose radius is. This circle is shown here: Sometimes we are given the equation for a circle in something other than standard form. If this is the case, we need to use a process called completing the square to turn it into standard form so that we can determine what the center and radius are. For eample, consider the following equation: 6 16 We can make the equation start to look "more standard" b recognizing that ( 0) and substituting: 6 ( 0) 16 But, unfortunatel, we still need to deal with the 's. If ou do something sneak (add 9 to both sides), it ends up working out well, because 6 9 factors nicel as ( )( ) or ( ) : 6 9 ( 0) ( ) ( ) ( 0) ( 0) 16 9 So, the equation was the equation of a circle with a center at (,0) and radius -- in disguise.

24 0- Equations of Circles / Obviousl, the hard part of the problem is knowing to add 9 to both sides. Fortunatel, it was not a luck guess. If ou multipl out ( N), ou get N N. So, the wa we got the 9 was b knowing that N 6 and therefore that N and therefore that N 9. Thus, we "completed the square" b adding 9 to 6. This method (completing the square) ma be new algebra for ou. When working with circles, it is also important to remember old algebra, including solving quadratic equations b factoring, b the quadratic formula, and b square roots. Consider the following problem: Eample 1 Find the -intercepts of the graph of the circle whose equation is 6 16.

25 Eample Consider the equation: 6 0- Equations of Circles /. Write this equation in standard form and then sketch a graph. Eample Complete the square: 1 Written Eercises A In problems 1-8 solve the equation b factoring ( )( ) ( ) 6. a a 6 7. b 6b

26 In problems 9-1 solve the equation b using the quadratic formula b b b 1. 8 In problems 1-18 solve the equation b completing the square z 6z c c d d 19. Consider the circle whose equation is ( ) ( ). a) Sketch the circle. b) Identif the coordinates of an -intercepts. c) Identif the coordinates of an -intercepts. 0. Consider the circle whose equation is ( ) ( ). a) Sketch the circle. b) Identif the coordinates of an -intercepts. c) Identif the coordinates of an -intercepts. In 1- find an equation for the circle shown Equations of Circles / 6 B In problems -6 solve the equation b completing the square Find (if an eist) the -intercepts of the circle whose equation is ( ) ( ) Find (if an eist) the -intercepts of the circle whose equation is ( ) ( ) 1.

27 0- Equations of Circles / 7 In problems 9- put the equations of the circles in standard form and then sketch them. 9. ( ) / In -6 find an equation for the circle shown.. 6. C 7. Sketch a graph of 8 9, labeling the circle s center and all intercepts. 8. Sketch a graph of ( ) 8 6, labeling the circle s center and all intercepts. If ou know three points, A, B, and C, on a circle, ou can find the circle s center b figuring out where the perpendicular bisectors of AB and BC intersect. In 9-0 find an equation for a circle assuming that points A, B, and C are on it. 9. A(,0), B(9,), and C(8,9) 0. A(6,), B(6,-1), and C(7,)

28 0-6 Problem Solving 0-6 Problem Solving / 8 Problem solving involves looking at a question and figuring out what tools are necessar to find the answer and how and in what order to use the tools. Often problem solving requires using multiple procedures that ou learned at different times and/or in a combination ou haven't done before. It is also often the case that all of the steps are eas, but figuring out what the steps are is a challenge. "We've never done a problem like this before," is a sill comment to make about problem solving that is precisel what makes it problem solving! Eample 1 A circle whose center is on the line 6 is tangent to the -ais at (,0). Find its equation. Eample Find the -intercept of the perpendicular bisector of the segment connecting (,6) and (6,1).

29 Eample Find all points on the line that are 6 units awa from (,0). 0-6 Problem Solving / 9

30 Written Eercises 0-6 Problem Solving / 0 A 1. Find k so that the slope between points (,) and (k,) is equal to /.. Find m so that the distance between (m, m) and the origin is equal to 8.. Find the -intercept of the perpendicular bisector of AB where A(,) and B(9,1).. Suppose A(,0) and B(-1,). Line l is perpendicular to AB, and l contains A. Find an equation for l.. Suppose A(7,1) and B(0,). Line l has slope and contains the midpoint of AB. Find an equation for l. 6. The line 1 is tangent to (barel touches) the circle with equation ( ) ( ). a) Sketch a graph of this situation. b) What is the slope of the tangent line? c) What is the point of tangenc (i.e. where do the line and the circle meet)? d) What is the slope of the line connecting the circle s center with the point of tangenc? e) What is the relationship between the slopes? 7. The line 9 is tangent to (barel touches) the circle with equation ( ) 17. a) Sketch a graph of this situation. b) What is the slope of the tangent line? c) What is the point of tangenc (i.e. where do the line and the circle meet)? d) What is the slope of the line connecting the circle s center with the point of tangenc? e) What is the relationship between the slopes? B 8. Suppose A(1,), B(n,), and C(,). Find n so that AB and BC are perpendicular. 9. Two different points on the line = are each eactl 1 units awa from the point (7,1). Sketch a graph of the situation and find the coordinates of the two points. 10. Two different points on the line = are each eactl units awa from the point (6,1). Sketch a graph of the situation and find the coordinates of the two points. 11. ( ) 10 Consider the sstem of equations. 6 a) Sketch graphs of both equations. b) Find the coordinates of both points of intersection. 1. ( ) 9 Consider the sstem of equations. ( ) a) Sketch graphs of both equations. b) Find the coordinates of both points of intersection.

31 0-6 Problem Solving / 1 1. One of the diameters of a circle is PQ where P(10,9) and Q(-6, -). Find an equation for the circle. 1. The line intersects the circle with equation ( ) ( ) 10 at two points, C and D. What is the distance between C and D? 1. The line intersects the circle with equation ( ) ( 9) at eactl one point. What is the distance between this point of intersection and the origin? C 16. Suppose A(0,). Find a point B in the first quadrant so that the slope of AB is / and the length of AB is a) Give three point that are the same distance from (,0) as the are from (7,0). b) There are infinitel man such points: Find an equation that describes them. 18. a) Give three point that are each 1 units from the point (,). b) There are infinitel man such points: Find the equation that describes them. 19. The length of a chord (a chord is a line segment where both endpoints are on a circle) is 1. If the distance from the midpoint of the chord to the center of the circle is 1, what is the radius of the circle? 0. A circle, whose center is a point on the line, is tangent to the -ais at (,0). Find an equation for the circle. In 1- make use of the following fact: A tangent line to a circle at point P is perpendicular to the segment connecting the center of the circle to the point of tangenc (point P). 1. Find an equation for the tangent line to a circle with center (,) if the point of tangenc is (,6).. Find an equation for the tangent line to a circle with center (-,) if the point of tangenc is (0,).