Module Four: Connecting Algebra and Geometry Through Coordinates

Transcription

1 NAME: Period: Module Four: Connecting Algebra and Geometry Through Coordinates Topic A: Rectangular and Triangular Regions Defined by Inequalities Lesson 1: Searching a Region in the Plane Lesson 2: Finding Systems of Inequalities That Describe Triangular and Rectangular Regions Lesson 3: Lines That Pass Through Regions Lesson 4: Preview of Topic B Topic B: Perpendicular and Parallel Lines in the Cartesian Plane Lesson 5: Criterion for Perpendicularity Lesson 6: Segments That Meet at Right Angles Lesson 7: Equations for Lines Using Normal Segments Lesson 8: Parallel and Perpendicular Lines Mid-Module Assessment Topic C: Perimeters and Areas of Polygonal Regions in the Cartesian Plane Lesson 9: Perimeter and Area of Triangles n the Cartesian Plane Lesson 10: Perimeter and Area of Polygonal Regions in the Cartesian Plane Lesson 11: Perimeters and Areas of Polygonal Regions Defined by Systems of Inequalities. Topic D: Partitioning and Extending Segments and Parameterization of Lines Lesson 12: Dividing Segments Proportionately Lesson 13: Analytic Proofs of Theorems Previously Proved by Synthetic Means Lesson 14: Motion Along a Line Search Robots again (Optional) Lesson 15: The Distance from a Point to a Line 1

2 Lesson 4: Parallel and Perpendicular Lines Classwork Opening Exercise Write the equation of the line that satisfies the following conditions: a. Has a slope of m = 1 4 and passes though the point (0, -5). b. Passes through the points (1, -3) and (-2, -1) Example 1 The line segment connecting A(3,1) to B(6,-4) is rotated counterclockwise 90 about the point A. a. Plot the points A, B. And then rotate them counterclockwise 90 about the point A. Label the new A, A and the new B, B. Draw segments AB and A B. b. What is the slope of AB and the slope of A B? c. How are the two segments related? How do their slopes compare? How could you use this to find B more directly.? 2

3 Exercise The point (a,b) is labeled to the right. a. Sketch a 90 counterclockwise rotation of (a,b) about the origin. What is the rotated location of (a,b) in terms of a and b? How are the two segments from the origin to the points related? What are their slopes? How are they related? How are the coordinates of the two points related? b. Sketch a 90 clockwise rotation of the point about the origin. What is the rotated location of (a,b) in terms of a and b? Find the slope of the line through the origin and (a, b) and the slope of the line through the origin and the new (a, b). How are the lines related? How are their slopes related? How are the coordinates of the points related? SUMMARY: ROTATION ABOUT THE ORIGIN CLOCKWISE 90 (a,b) becomes (, ) ROTATION ABOUT THE ORIGIN COUNTERCLOCKWISE 90 (a, b) becomes ROTATION ABOUT A POINT OTHER THAN THE ORIGIN 90, we can use the negative reciprocal of the slope to find the new point. (, ) 3

4 Problem Set 1. Find the new coordinates of point (0,4) if it rotates a. 90 counterclockwise b. 90 clockwise c. 180 counterclockwise d. 270 clockwise 2. What are the new coordinates of the point (-3, -4) if it is rotated about the origin: (Can you do this using the pattern we saw with the coordinates?) a. Counterclockwise 90? b. Clockwise Line segment ST connects points S(2,1) and T(6,4). (Can you do this using the slopes?) a. Where does point T land if the segment is rotated 90 counterclockwise about S? b. Where does point T land if the segment is rotated 90 clockwise about S? c. What is the slope of the original segment? d. What is the slope of the rotated segments? 4

5 4. Line segment VW connects points V(0,0) and W (4, -3). a. Where does point W land if the segment is rotated 90 counterclockwise about V? b. Where does point W land if the segment is rotated 90 clockwise about V? c. Where does point V land if the segment is rotated 90 counterclockwise about W? d. Where does point V land if the segment is rotated 90 clockwise about W? e. Does it make sense with your sketch? 5. If the slope of a line is 0, what is the slope of a line perpendicular to it? IF the line has lope 1, what is the slope of a line perpendicular to it? 6. If a line through the origin has a slope of 2, what is the slope of the line through the origin that is perpendicular to it? 7. A line through the origin has a slope of 1. Carlos thinks the slope of a perpendicular line at the 3 origin will be 3. Do you agree? Explain why or why not? 8. Could a line through the origin perpendicular to a line through the origin with the slope ½ pass through the point (-1, 4)? Explain how you know. 5

6 Resource Page Pythagorean Theorem The sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse. Distance Formula - Properties of Parallelograms Opposite angles are. Consecutive angles are. Opposite sides are and. Diagonals each other. If one angle is a right angle, then. 6

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8 Lesson 5: Criterion for Perpendicularity Entry Task 1) In the right triangle ABC, find the missing side measure. a. If AC = 7 and AB = 20. Leave answers in simplified radical form. b. If AC = CB and AB = 2, what is AC (and CB)? Leave answers in simplified radical form. c. 1) Properties of parallelograms: In parallelograms (fill in the blanks) a. Opposite angles are. b. Consecutive angles are. c. Opposite sides are and. d. Diagonals each other. e. If one angle is a right angle, then. 8

9 Example 1: Is a triangle with the following side lengths a right triangle: AC = 4, AB = 7, BC = 5. If so, name the right angle. A teacher asks a student to determine if a triangle with sides 5, 5 2 and 5 is a right triangle. The student states that the triangle is not a right triangle because (5 2) Is the student correct? Explain. Example 2: a. Plot the points O(0, 0), A(6, 4) and B(-2, 3) on the coordinate plane. b. Do you think ABO is a right triangle? How could we prove it? c. If ABO is a right triangle, which would be the right angle? d. What is the relationship between the legs of a right triangle? 9

10 Example 3: a. Plot the points O(0, 0), P(3, -1) and Q(2, 3) on the coordinate plane. Determine whether OP and OQ are perpendicular. Show work to support your conclusion. Example 4: a. Referring to the diagram below, what must be true if OA OB? b. Does this generalization apply to the points found in Example 2? 10

11 Theorem: What is the converse of this theorem? Is the converse true as well? How do you know? 11

12 Example 5: Theorem: Two non-vertical lines, l 1 and l 2 are perpendicular iff (if and only if) their slopes are negative reciprocals of each other. Let l 1 and l 2 be two non-vertical lines, such that both pass through the origin. Using the figure to the right, fill in the reasons for the proof of the above theorem. Statement 1) l 1 l 2 l 1 is defined by the equation y = m 1 x and l 2 is defined by the equation y = m 2 x Points (0, 0) and (1, m 1 ) are on l 1 Points (0, 0) and (1, m 2 ) are on l 2 2) m 1 m 2 = 0 Reason 1) 2) 3) 1 + m 1 m 2 = 0 3) 4) m 1 m 2 = 1 4) 5) m 1 = 1 m 2 5) What does this show us about the product of the slopes of perpendicular lines? Does this apply to all perpendicular lines? 12

13 Exercises: 1) The points O(0, 0), A(-4, 1), B (-3, 5) and C(1, 4) are the vertices of a parallelogram OABC. Is this parallelogram a rectangle? Provide mathematical support for your answer. You may use the attached graph paper to sketch the problem. 2) Prove, using the Pythagorean Theorem, that AC AB, given A(-2, -2), B(5, -2) and C(-2, 22). Show work and explain your rational in words. 3) Given points O(0, 0), S(2, 7) and T(7, -2), where OS OT, will the images of the segments be perpendicular if the three points O, S, and T are translated four units to the right and eight units up? Show work and explain your rational in words. 4) Jimmy thinks the points (4, 2) and (-1, 4) form a line perpendicular to a line with a slope of m = 4. Do you agree? Why or why not? 13

14 Lesson 6: Segments that Meet at Right Angles Entry Task: 1) Sida thinks that the segment having endpoints A(0, 0) and B(6, 0) is perpendicular to the segment with endpoints A(0, 0) and C(-2, 0). Do you agree? Why or why not? 2) Given A(0, 0) and B(3, -2), find the coordinates of a point C so that AC AB. You may work with 1 partner on this problem. 3) Properties of a Kite: In kites (fill in the blanks) a. Exactly one pair of opposite angles are. b. Two pairs of sides are. c. Diagonals are. Examples 1-4: Group Work 1) Given AB CD and A(3, 2), B(7, 10), C(-2, -3), and D(4, d 2 ), find the value of d 2. Use the attached graph paper as an aid and show all work to justify your answer. 2) A triangle in the coordinate plane has vertices A(0, 10), B(-8, 8) and C(-3, 5). Is this triangle a right triangle? If so, which vertex is the right angle? How do you know? 14

15 3) A(-7, 1), B(-1, 3), C(5, -5) and D(-5, -5) are vertices of a quadrilateral. If AC bisects BD, but BD does not bisect AC, determine whether ABCD is a kite. Show all work to justify your answer. 4) Given points S(2, 4), T(7, 6), U(-3, -4) and V(-1, -9), are the lines ST Justify your response. and UV perpendicular? Exercises: You may use the attached graph paper to help solve these problems. 1) Given point O is at the origin, determine if AO BO, for the following points A and B. a. A(9, 10), B(10, 9) b. A(9, 6), B(4, -6) 2) Given M(5, 2), N(1, -4) and L has the coordinates listed below, determine if LM. MN a. L(-1, 6) b. L(11, -2) c. L(9, 8) 15

16 3) A quadrilateral has vertices A(2 + 2, 1), B(8 + 2, 3), C(6 + 2, 6) and D( 2, 2). Prove the quadrilateral is a rectangle. 4) A robot begins at position (-80, 45) and moves on a path to (100, -60). It turns 90⁰ counterclockwise. a. Find the x-coordinate of the point on its path having the y-coordinate 120. b. Write an equation of the line of motion after its 90⁰ turn. c. If the robot stops to charge on the x-axis, what is the exact location where it will stop? 5) A right triangle has vertices (1, 3) and (6, -1) and a third vertex located in Quadrant IV. a. Determine the coordinates of the missing vertex. b. Reflect the triangle across the y-axis. What are the new vertices? c. If the original triangle is rotated 90⁰ counter clockwise about (6, -1). What are the coordinates of the other vertices? What do you notice? 16

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18 Lesson 7: Equations for Lines Using Normal Segments Entry Task: Rewrite each of these equations of a line, given in standard form, as slope-intercept form. Then identify the slope and y-intercept of each. 1) 6x + 3y = 12 2) 5x +7y = 14 3) 2x 5y = -7 Example 1: Given point A(a, b) which lies on line l, point B(c, d) not on line l, and AB l: a. Draw a diagram, in Quadrant I, illustrating this information. We call segment AB a normal segment to line l. Definition: Normal Segment to a Line 18

19 Example 2: Given A(5, -7) and B(8, 2): a. Find equation for the normal line, going through A. b. Find equation for the normal line, going through B. Example 3: Given U(-4, -1) and V(7, 1): a. Find an equation for a line through U perpendicular to UV. b. Find an equation for a line through V perpendicular to UV. 19

20 Exercises: 1) Given points C(-4, 3) and D(3, 3): a. Find the equation for the normal line, passing through point C. b. Find the equation for the normal line, passing through point D. 2) Given points N(7, 6) and M(7, -2): a. Find the equation for the normal line, passing through point N. b. Find the equation for the normal line, passing through point M. 20

21 3) A line segment has endpoints (2, 50) and (20, 65). a. Graph the line segment on the attached graph paper. b. Find the equations of the lines perpendicular to this segment, passing through each end point (note: this means two different line equations, not one line passing through both end points). c. Graph the two lines you found in (b). d. What do you notice about these two lines? 4) Write an equation of a line that passes through the origin that intersects the line 2x +5y = 7 to form a right angle. 21

22 Lesson 8: Parallel and Perpendicular Lines Entry Task 1) Determine whether the lines given by the equations 2x + 3y = 6 and y = 3 x + 4 are 2 perpendicular. 2) Two lines having the same y-intercept are perpendicular. If the equation of one of these lines is y = 4 x + 6, what is the equation of the second line? 5 Example 1: a. Using a straight edge, construct a line and label it k. b. Using your compass and straight edge, construct a line l 1 perpendicular to k. c. Using your compass and straight edge, construct a second line, l 2 perpendicular to k. d. What is the relationship between lines l 1 and l 2? What theorems or postulates can be used to prove your conclusion? e. Write a generalized conditional statement describing the relationship between two coplanar lines that are perpendicular to the same line. f. If the slope of line k is m, what is the slope of line l 1? 22

23 g. If the slope of line k is m, what is the slope of line l 2? h. What do you notice about the slopes of lines l 1 and l 2? Theorem: Examples 2-6: Group Work 2) Given the line y = 2x 8: a. What is the slope of the line? b. What is the slope of any line parallel to the given line? c. Write an equation of a line passing through point (-3, 6) and parallel to the line. d. What is the slope of any line perpendicular to the give line? Explain. 3) Find the equation of a line through (0, -7) and parallel to the line y = 1 x a. If a line is perpendicular to y = 1 x + 5, will it be perpendicular to the line you just 2 found? 23

24 4) Find the equation of a line through ( 3, 1 ) and parallel to: 2 a. x = -9 b. y = 7 c. What is the relationship between the lines you found in (a) and (b)? d. What generalized conjecture can we make about all vertical lines? What generalized conjecture can be make about all horizontal lines? 5) Find an equation of a line through ( 2, π), parallel to the line x 7y = 5. 6) A robot moves at a constant speed of 2 feet/second, starting at position (20,50) on the coordinate plane. The robot moves in a south-east direction along the line 3x + 4y = 260. After 15 seconds it turns left 90⁰ and travels in a straight line in this new direction. Use the attached graph paper to solve the following problems. a. What are the coordinates of the point at which the robot made the turn? Explain how you found this point. 24

25 b. Find an equation for the second line on which the robot traveled. c. If, after turning, the robot travels for 20 seconds along this line and then stops, how far will it be from its starting position? d. What is the equation of the line the robot needs to travel along in order to return to is starting position? e. How long will it take for the robot to get back to the starting position? 25

26 Exercises: 1) Are the pairs of lines parallel, perpendicular, or neither. Justify your response mathematically. a. 3x + 2y = 74 and 9x 6y = 15 b. 4x 9y = 8 and 18x + 8y = 7 2) Write an equation of a line through ( 3, 5 ) and: (leave all answers in radical/fraction format) 4 a. Parallel to y = 7 b. Perpendicular to y = 7 c. Parallel to 1 x 3 y = d. Perpendicular to 1 x 3 y = ) A vacuum robot is in a room and charging at position (0, 5). Once charged it begins to move on a north-east path at a constant speed of 1 foot/second along the line 4x 3y = -15. After 60 2 seconds it turns right 90⁰ and travels in the new direction. a. What are the coordinates of the point at which the robot made the turn? 26

27 b. Find an equation for the second line on which the robot traveled. c. If after turning, the robot travels for 80 seconds along this line, how far has it traveled total (from the starting position)? d. What is the equation of the line the robot needs to travel along in order to return to its charging station? How long will it take to get there? 4) Based on the figure below, is AB DE? Explain. 27