GEOMETRY HONORS COORDINATE GEOMETRY PACKET

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1 GEOMETRY HONORS COORDINATE GEOMETRY PACKET Name Period 1

2 Day 1 - Directed Line Segments DO NOW Distance formula D x x y y Midpoint formula x x, y y 2 2 M Slope formula y y m x x Equation of a line y mx b (Slope-Intercept Form) y - y 1 = m(x x 1 ) (Point-Slope Form) 1) What is the slope of the line passing through the points 8,2? 2,5 and 2) If M 3,4 is the midpoint of AB and the coordinates of A are of B. 9,8, then find the coordinates 3) Graph the line that passes through 3, 2 and has a slope of ) Given points C and D, find the distance of CD: C 6, 2 D 4, 8 What is the equation of this line? 2

3 A directed line segment is a segment that has distance and a direction. This definition basically defines a segment to be a vector - magnitude and direction. The directed line segment AB implies that we are starting at point A and going towards point B. This means that the initial point is A. This is important because when we partition (or divide up) the segment into a ratio, the ratio will reference a starting point and a direction. For example, partitioning directed line segment AB into a 1:3 ratio implies that we start at point A and then have 1 part followed by 3 parts for a total of 4 parts. A ratio of 3:1 would place the three parts first starting at point A then then the 1 part would follow. This results in two very different partitions. The diagrams below demonstrate these two different ratios. Given the directed line segment AB, determine point P so that it divides the segment into the ratio of: A 1 P 1:3 3:1 The ratio is 1:3 but the segment has 4 parts. 3 B A 3 The ratio is 3:1 but the segment has 4 parts. P 1 B We can reference the same partition of a line segment by using the different endpoints of the directed segment. The diagram below demonstrates how you can reference the same location using either endpoint of the line segment. Note how the wording changes for these two descriptions. A Directed line segment AB is divided into a ratio of 1:2 1 P 2 B A Directed line segment BA is divided into a ratio of 2:1 1 P 2 B The ratio is 1:2 but the segment has 3 parts. The ratio is 2:1 but the segment has 3 parts. Example 1: Determine the ratio of the directed line segment BA when partitioned by point P. (Hint: B is the initial point) B P A B P A A P B a) : b) : c) : 3

4 Partitioning a Directed Line Segment Partitioning a line segment means to divide it up into pieces. To relate this to a dilation it means that we will be doing a reduction (0 < k < 1) so that the point will be on the segment. A is the initial point, our center of dilation. D 1 ( B) (4,2) D 1 ( B) (2,1) A, A,4 2 B 1 1 B 1 3 B A A Partitioning AB into a 1:1 ratio. A Partitioning AB into a 1:3 ratio. When we dilate by a value between 0 and 1 we place the image of our point onto the segment. This point partitions the segment into two parts. Be careful, the scale factor is not exactly the partitioning ratio but they definitely have a strong relationship. A scale factor represents the amount out of the total while the ratio is a comparison of the two pieces. A little practice converts these two value back and forth easily. Partition Ratio Dilation Scale Factor Partition Ratio Dilation Scale Factor Partition Ratio Dilation Scale Factor Partition Ratio 1:2 2:1 3:4 4:5 Dilation Scale Factor If we move the initial point (the center of dilation) then we adjust our relationship to represent that change. D (a,b),k = (k x + a, k y + b) OR Step 1: Find Horizontal Distance (run) = x 2 x 1 = x Step 2: Find Vertical Distance (rise) = y 2 y 1 = y Step 3: Find Dilation Scale Factor = k = part whole Step 4: D (a,b),k = (k x + a, k y + b) 4

5 Ex. #2 - Determine the point P that partitions the directed line segment AB into a ratio of 1:1, where A (-5,2) and B (3,6). Ex. #3 - Determine the point P that partitions the directed line segment AB into a ratio of 2:3, where A (1,-5) and B (9,-1). 5

6 Day 1 - HW 6

7 7

8 8

9 Day 2 - Writing Equations of a Line Warm Up 1. The dilation, D 3 ( B) B' partitions AB into a ratio AB :B B of: A,4 A. 3:4 B. 1:3 C. 3:1 D. 3:7 2. Determine the ratio of the directed line segment AB when partitioned by point P. (Hint: A is the initial point) B P A A P B A P B a) : b) : c) : 3. Given directed line segment AB, where A(0,0) and B(18,0). Determine the point P that partitions the segment into a 5:1 ratio. A. P(3,0) B. P(6,0) C. P(12,0) D. P(15,0) 9

10 Write an equation of a line that is parallel to the line 2x + y = 6 and whose y-intercept is the same as the line y = x - 2. What do you know about slopes of parallel lines? Method 1: Slope Intercept Form Formula: Method 2: Point Slope Form Formula: Answer: Answer: 1. Write an equation of a line perpendicular to y = 3 1 x - 6 and has a y-intercept equal to 10. What do you know about slopes of perpendicular lines? Method 1: Slope Intercept Form Formula: Method 2: Point Slope Form Formula: Answer: Answer: 10

11 You try It! 3. Write an equation of a line that passes through the point (1, 5) and is perpendicular to 2y = x - 6. Answer: 4. Write an equation of a line that is parallel to the line 6x 2y = 14 with an x-intercept of 5. Answer: 11

12 Example 5: Write the equation of a line passing through the two points given. (10, 20) and (20, 65) Step 1: Calculate Slope - Step 2: Use your calculate Slope to help you write your equation. Method 1: Slope Intercept Form Method 2: Point Slope Form Formula: Formula: Answer: Answer: Practice: Write the equation of a line passing through the two points given. (2, 5) and ( 8, 5) Answer: 12

13 Horizontal and Vertical lines Slope Write the equation of the horizontal line and/or vertical passing through each point. 6. (3, 7) 7. (2, -4) Horizontal line: Horizontal line: Vertical Line: Vertical Line: Answer: 10. Write an equation of a line that is parallel to the y axis and contains the point ( 6, 1). Answer: 11. Write an equation of a line that is parallel to the x axis and contains the point (7, 1 3 ). Answer: Answer: 13

14 SUMMARY Memory Device for Vertical Lines Memory Device for Horizontal Lines x = a value of point = (a, b) y = b value of point = (a, b) Exit Ticket 1) 2) 14

15 Day 2 Homework 15

16 8. Write (If possible, in point-slope form) an equation of the line. a) Containing (2, 1) and (3, 4) b) Containing (-6, 3) and (2, -1) c) Containing (1, 5) and (-3, 5) d) With an x-intercept of 2 and a slope of 7 e) That has an x-intercept of 3 and passes through (1, 8) f) That passes through (-3, 6) and (-3, 10) g) That passes through (8, 7) and is perpendicular to the graph of 3y = -2x

17 9. The line that represents the equation y = 8x 1 contains the point (k, 5). Find k. 11. Determine the point P that partitions AB, A(-6,8) B(1,-20) into a 3:4 ratio. 17

18 Day 3 - Writing Equations of Altitudes, Medians and Perpendicular Bisectors Warm - Up Determine the point P that partitions AB, A(1,2) B(11,22) into a 1:4 ratio. A. (3,6) B. (5, 10) C. (7, 14) D. (9, 18) 18

19 A median of a triangle is a line segment drawn from the vertex of a triangle to the midpoint of the opposite side. How to calculate the median AD to side BC of ABC A = (4, 10), B = (12, 6), and C = (8, 2) Formula: Formula: Method 1: Slope Intercept Form Formula: Method 2: Point Slope Form Formula: Answer: Answer: 19

20 An altitude of a triangle is a line segment drawn from a vertex of a triangle perpendicular to the opposite side. How to calculate the altitude AD to side BC of ABC A = (4, 10), B = (12, 6), and C = (8, 2) Formula: Formula: Formula: Method 1: Slope Intercept Form Formula: Method 2: Point Slope Form Formula: Answer: Answer: 20

21 A Perpendicular bisector of a line segment is a line (or line segment) that is perpendicular to the segment at its midpoint. How to calculate the bisector AD to side BC of ABC A = (4, 10), B = (12, 6), and C = (8, 2) Method 1: Slope Intercept Form Formula: Method 2: Point Slope Form Formula: Answer: Answer: 21

22 SUMMARY Exit Ticket 22

23 In problems 13-17, use ABC in the diagram. Day 3 Homework 13. Write, in point-slope form, an equation of a line through C parallel to AB. 14. Write an equation of the perpendicular bisector of AB. 15. Write an equation of the altitude from C to AB. 16. Write an equation of the median form C to AB. 17. Find the slope of the line passing through the midpoints of AC and BC. 23

24 18. A line passes through a point 3 units to the left of and 2 units above the origin. Write an equation of the line if it is parallel to a) The x-axis b) The y-axis 20. Determine the point P that partitions AB, A(-3,-14) B(21,2) into a 3:5 ratio. 24

25 Circles in the Coordinate Plane Day 4 SWBAT: Write equations and graph circles in the coordinate plane. Warm Up Find the value that completes the square and then rewrite as a perfect square. 25

Example 1: Writing the Equation of a Circle Model Problem: J with center J (2, 2) and radius 4 Practice #1: Write the equation of circle L with center L ( 5, 6) and radius 9.

26 Example 1: Writing the Equation of a Circle Model Problem: J with center J (2, 2) and radius 4 Practice #1: Write the equation of circle L with center L ( 5, 6) and radius 9. Example 2: Identifying the center and radius from the equation of a circle. Model Problem: Example 3: Identifying the center and radius from the graph of a circle. Model Problem: Find the center and radius of the circle, and then write its equation. 26

27 Example 4: Writing the Equation of a Circle given a Center and a Point on the Circle P with center P(0, 3) and passes through point (6, 5). Example 5: Writing the Equation of a Circle given Two Endpoints. Model Problem: Writing the equation of K that passes through endpoints A(5, 4) and B(1, 8). Step 1: Calculate the Midpoint. Step 2: Calculate the radius of the circle. Step 3: Plug in Midpoint, and radius into circle equation. Practice #2: Write the equation of circle Q that passes through (2, 3) and (2, 1) 27

28 Example 6: Write the equation of the circle from General Form to Standard Form. Write the equation of the circle and identify the center and radius. Practice #3: Write the equation of the circle and identify the center and radius. 28

29 Example 7: Graphing Circles Practice #6 Graph: x 2 + y 2 = 16. Graph: (x +5) 2 + (y - 2) 2 = 4. Challenge 29

30 SUMMARY Exit Ticket

31 Homework Day 4 31

32 32

33 A circle passes through 2 points (2, 2) and (4, 2). Write the equation of the circle. 33

34 7) 8) 9) 34

35 Day 5: System of Equations SWBAT: Graph the Solutions to Quadratic Linear Systems Warm Up Algebra 35

36 Example: 2. 36

37 Quadratic Linear System of Equations Quadratic Equation Linear Equation m = y = x + 3 Vertex = (, ) b = 37

38 2. Quadratic Equation Linear Equation Vertex = (, ) m = b = 38

40 Summary Exit Ticket 40

41 1. Day 5 HW 2. Graph the system and find the points of intersection

44 Day 6: Writing Equations of a Line Using Points of Intersection 1. Write the equation of a line that contains the point of intersection of the graphs x = 4 and y = 2x + 8 and is parallel to the line whose equation is y = -2x + 5. Step 1: Solve the system of equations to determine the point of intersection. Step 2: Write the equation of your line using the point of intersection from above. Method 1: Slope Intercept Form Formula: Method 2: Point Slope Form Formula: Answer: Answer: 44

45 2. Write the equation of a line that contains the point of intersection of the graphs 8x 3y = 7 and 1 10x + 4y = -1 and is perpendicular to the line y x 7. 3 Step 1: Solve the system of equations to determine the point of intersection. Step 2: Write the equation of your line using the point of intersection from above. Method 1: Slope Intercept Form Formula: Method 2: Point Slope Form Formula: Answer: Answer: 45

46 Using Equations of Lines to Find the Coordinates of an Altitude 3. In ABC with coordinates A(-3,4), B(6,-2) and C(7,6) altitude CD is drawn. Find the coordinates of D. Step 1: Calculate the equation of the line where the altitude intersects the side. Step 2: Calculate the equation of the altitude. Step 3: Solve the systems from above for D 46

47 You try! a) In ABC with coordinates A(0,0), B(6,3) and C(1,5) altitude CD is drawn. Find the coordinates of D. Ans: D = ( 14 5, 7 5 ) 47

48 b) In ABC with coordinates A ,, B 4,4 2 2, and C 4,3 altitude BD is drawn, find 2 the coordinates of D. Ans: D = ( , ) 48

49 c) In ABD with coordinates A(-4,1), B(1,5) and C(6, -1) altitude CD is drawn. Find the coordinates of D. Ans: D = (1.1219, ) 49

50 Day 6 - Homework 7) Show that the graphs of the following 3 equations are concurrent (intersect at a single point). What are the coordinates of the point of intersections? 2x + 3y = 2 { y = 2x 10 3x y = 14 9) Find, in point-slope form, an equation of the line containing (2, 1) and the point of intersection of the graphs of 3x y = 3 and x + 2y =

51 10) Find an equation of the line that is parallel to the graph of 2x + 3y = 5 and contains the point of intersection of the graphs of y = 4x + 8 and y = x

52 13) In ABD with coordinates A(-4,1), B(1,5) and C(6, -1) altitude CD is drawn. Find the coordinates of D. 52

53 15) Find the distance between the parallel lines corresponding to y = 2x + 3 and y = 2x + 7. (Hint: Start by choosing a convenient point on one of the lines.) 53

54 54

55 4. The vertices of triangle PQR are P(1,2), Q(-3,6) and R(4, 8). a. Find the coordinates of S, the midpoint of PQ. b. Express in radical form, the length of the median RS. c. Find the slope of PR. d. A line through point Q is parallel to PR. If the line passes through the point (x, 14), find the value of x. 5. Find the coordinates of the midpoint of CD: C , 2 D , Find the distance from S to T: S (x + y, a + b) T (x y, b - a) 7. Simplify: a) 3 50 b) 2 12 c)

56 8. In ABD with coordinates A(-4,1), B(1,5), C(6, -1) altitude CD is drawn. Find the coordinates of D. 9. Write an equation of the perpendicular bisector of the segment that joins the points (3, -7 ) and (5,1). 10. Write an equation of a line that passes through the point B(3,1) and is perpendicular to the line 3y + 2x =

57 11. The vertices of ABC are A(0,6), B( -8, 0), C(0,0). Write an equation of the line that passes through one of the vertices of the triangle and parallel to AC. 12. Write an equation of the line that contains point (-5,2) and is parallel to the y axis. 13. Write an equation of the line that contains point (4,-1) and is perpendicular to the x axis. 14. Write an equation of a line that contains the point (2, 2) and the intersection of the graphs x + y = 10 and x y = 2. 57

58 15. In ABC with coordinates A(-4,3), B(2,7) and C(4,-3). a) Find the equation of the median to AC. b) Find the equation of the altitude to AB. c) Find the equation of the perpendicular bisector of AB. 58

59 16. The dilation, D 3 ( B) B' partitions AB into a ratio AB :B B of: A,4 A. 3:4 B. 1:3 C. 3:1 D. 3:7 17. Directed line segment AB is partitioned by point P into a ratio of 4:1. Which of the following represent this relationship? A. B. B P A A P B C. D. A P B A P B 18. Determine the point P that partitions AB, A(-3,-14) B(21,2) into a 3:5 ratio. 59

60 19. Which of the following is true about the equation: x 2 + (y - 8) 2 = 3? a) The center is at the origin and the radius is 3. b) The center is at (0,8) and the radius is 3. c) The center is at (0,8) and the radius is 9. d) The center is at (0,8) and the radius is Given the equation: x 2 + y 2 + 6x - 6y + 6 = 0 a) Write each equation in center-radius form. b) Find the coordinates of the center. c) Find the radius of the circle. d) Graph the circle. 21. Write the equation of a circle given center (6, 5) and the point on the circle (0, -3). Then graph. 60

61 22. Part a: Find the points of intersection of the line y = 2x + 1 and the circle x 2 + y 2-2y 4 = 0 Part b: Show that the line y = 2x + 1 is a diameter of the circle. 61

62 Answer Key: 1. (-16,16) a) b) 3 3. k = a) (-1,4) b) 41 c) 2 d) x = , y 4a 2 y a 7. a) 15 2 b) 4 3 c equation of AB : y 1 = 4 (x + 4); 5 y 4 21 x 5 5 equation of altitude: y + 1 = - 5 (x 6); y = x 4 2 coordinates of D: 46, y + 3 = (x 4) or y = x y 1 = 3 (x 3) or y = 3 x x = x = x = y 2 = 1 (x 2) or y 4 = 1 (x 6) or y = 1 x a) y 0 = 7 2 (x 0) or y = 7 2 x b) y + 3 = 3 2 (x 4) or y = x + 3 c) y - 5 = 3 (x + 1) or y = - 3 x C 17. D 18. (6,-8) 62

63 19. D

64 Additional Questions: Honors Text Book Pages: : 13, 17, 21ab, 29 13) 17) or y 4 = 2(x 7) 21a) 21b) 64

GEOMETRY HONORS COORDINATE GEOMETRY PACKET

GEOMETRY HONORS COORDINATE GEOMETRY PACKET Name Period Homework Lesson Assignment Day 1 - Slopes of Perpendicular WKSHT and Parallel Lines Day 2 - Writing an Equation of a Line HW- Honors TXTBK pages 615-617