25.4 Coordinate Proof Using Distance with Quadrilaterals

Transcription

1 - - a a 6 Locker LESSON 5. Coordinate Proof Using Distance with Quadrilaterals Name Class Date 5. Coordinate Proof Using Distance with Quadrilaterals Essential Question: How can ou use slope and the distance formula in coordinate proofs? Resource Locker Common Core Math Standards The student is epected to: G-CO.11 Prove theorems about parallelograms. Also G-GPE., G-GPE.5 Mathematical Practices MP.6 Precision Language Objective Eplain in our own words how to prove that a quadrilateral on a coordinate plane is a rectangle. ENGAGE Essential Question: How can ou use slope and the distance formula in coordinate proofs? You can use the slope to show that opposite sides of a quadrilateral have the same slope and are parallel. You can also use slope to show that adjacent sides of a quadrilateral are perpendicular if the product of their slopes is -1. You can use the distance formula to find the lengths of diagonals and sides of quadrilaterals to show congruence. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo. Connect the photo to the Lesson Performance Task b mentioning that the task will show how orderl congruent line segments can be found inside a figure composed of disorderl line segments of different lengths. Then preview the Lesson Performance Task. Houghton Mifflin Harcourt Publishing Compan Eplore Positioning a Quadrilateral on the Coordinate Plane You have used coordinate geometr to find the midpoint of a line segment and to find the distance between two points. Coordinate geometr can also be used to prove conjectures. Remember that in Lesson 10.3 ou learned several strategies that make using a coordinate proof simpler. The are: Use the origin as a verte, keeping the figure in Quadrant I. Center the figure at the origin. Center a side of the figure at the origin. Use one or both aes as sides of the figure. Position a rectangle with a length of 8 units and a width of 3 units in the coordinate plane as described. A B Method 1 Center the longer side of the rectangle at the origin. Method Use the origin as a verte of the rectangle. Depending on what ou are using the figure to prove, one method ma be better than the other. For eample, if ou need to find the midpoint of the longer side, use the first method. A coordinate proof can also be used to prove that a certain relationship is alwas true. You can prove that a statement is true for all right triangles without knowing the side lengths. To do this, assign variables as the coordinates of the vertices. Position a square, with side lengths a, on a coordinate plane and give the coordinates of each verte. C Sketch the square. Label the side lengths. Possible answer: Position the square so that one side is on the -ais and another side is on the -ais. (-, 3) (-, 0) - - (0, 3) (, 3) (, 0) 0 (8, 3) (0, 0) (8, 0) Module Lesson Name Class Date 5. Coordinate Proof Using Distance with Quadrilaterals Essential Question: How can ou use slope and the distance formula in coordinate proofs? G-CO.11 Prove theorems about parallelograms. Also G-GPE., G-GPE.5 Houghton Mifflin Harcourt Publishing Compan Eplore Positioning a Quadrilateral on the Coordinate Plane You have used coordinate geometr to find the midpoint of a line segment and to find the distance between two points. Coordinate geometr can also be used to prove conjectures. Remember that in Lesson 10.3 ou learned several strategies that make using a coordinate proof simpler. The are: Use the origin as a verte, keeping the figure in Quadrant I. Center the figure at the origin. Center a side of the figure at the origin. Use one or both aes as sides of the figure. Position a rectangle with a length of 8 units and a width of 3 units in the coordinate plane as described. Method 1 Center the longer side of the rectangle at the origin. Method Use the origin as a verte of the rectangle. Depending on what ou are using the figure to prove, one method ma be better than the other. For eample, if ou need to find the midpoint of the longer side, use the first method. (0, 3) (8, 3) (0, 0) (8, 0) 0 8 A coordinate proof can also be used to prove that a certain relationship is alwas true. You can prove that a statement is true for all right triangles without knowing the side lengths. To do this, assign variables as the coordinates of the vertices. Position a square, with side lengths a, on a coordinate plane and give the coordinates of each verte. Sketch the square. Label the side lengths. Possible answer: Position the square so that one side is on the -ais and another side is on the -ais. (-, 3) Resource (, 3) (-, 0) (, 0) 0 Module Lesson a a HARDCOVER PAGES Turn to these pages to find this lesson in the hardcover student edition Lesson 5.

2 What are the coordinates of each verte? The vertices are (0, 0), (0, a), (a, a), and (a, 0). Reflect (0, a) a (0, 0) (a, a) a (a, 0) EXPLORE Positioning a Quadrilateral on the Coordinate Plane 1. Discussion Describe another wa ou could have positioned the square and give the coordinates of its vertices. Possible answer: You could have positioned the square so that (0, 0) was the midpoint of. When writing a coordinate proof wh are variables used instead of numbers as coordinates for the vertices of a figure? Possible answer: B using variables, our results are not limited to specific numbers. Eplain 1 Proving Properties of a Parallelogram You have alread used the Distance Formula and the Midpoint Formula in coordinate proofs. As ou will see, slope is useful in coordinate proofs whenever ou need to show that lines are parallel or perpendicular. Eample 1 one side. Then the coordinates of its vertices would be (-a, 0), (-a, a), (a, a), and (a, 0). Prove or disprove that the quadrilateral determined b the points A (, ), B (3, 1), C (, 1), and D ( 1, ) is a parallelogram. Use slopes to write the coordinate proof. To determine whether ABCD is a parallelogram, find the slope of each side of the quadrilateral. Slope of AB - 1 = - = = -3-1 = 3; Slope of BC - 1 = - = = - -5 = 5 ; Slope of CD - 1 = - = - (-1) (-) = 3 1 = 3; Slope of DA - 1 = - = (-1) = 5 Compare slopes. The slopes of opposite sides are equal. This means opposite sides are parallel. So, quadrilateral ABCD is a parallelogram. - C D A B 0 Houghton Mifflin Harcourt Publishing Compan INTEGRATE TECHNOLOGY Students have the option of completing the Eplore activit either in the book or online. QUESTIONING STRATEGIES Are rectangles the onl quadrilaterals that ou can position with a side on each of the aes? No, ou can position an quadrilateral that contains at least one right angle so that two of its sides are on an ais. EXPLAIN 1 Proving Properties of a Parallelogram AVOID COMMON ERRORS Advise students to pa close attention to the markings on a diagram, especiall when writing a proof. Some quadrilaterals have onl one set of parallel lines; these are not parallelograms. Module Lesson PROFESSIONAL DEVELOPMENT Math Background Analtic geometr is the stud of geometr using a coordinate sstem. Although certain aspects of the subject date to ancient times, the French mathematician René Descartes ( ) is traditionall considered the father of analtic geometr. The strength of analtic geometr lies in its use of tools from algebra to solve geometr problems. In particular, this lesson shows how it is possible to use a Cartesian coordinate sstem to classif quadrilaterals. Coordinate Proof Using Distance with Quadrilaterals 1308

3 QUESTIONING STRATEGIES How could ou use triangles in tring to prove properties of quadrilaterals? You can divide a quadrilateral into triangles and then use triangle congruence to show that angles, sides, or diagonals of a quadrilateral are congruent. B Use the Distance Formula to write the coordinate proof. To determine whether ABCD is a parallelogram, find the length of each side of the quadrilateral. Remember that the Distance Formula is length = ( - 1 ) + ( - 1 ). AB = ( - ) + (1 - ) BC = 3 (- - 3 ) -1 = (-1) + ( ) = -3 (-5) + ( - ) = = ( - 1) CD = ( ) = (1) + ( 3 ) = 10 + ( - (-1)) DA = ( - -1 ) + ( - ) = ( 5 ) + ( ) = 9 Compare the side lengths. The lengths of the opposite sides are equal. B the Opposites Sides Criterion for a Parallelogram, we can conclude that ABCD is a parallelogram. Reflect 3. Suppose ou want to prove that a general parallelogram WXYZ has diagonals that bisect each other. Wh is it convenient to use general verte coefficients, such as a and b? To prove that the diagonals bisect each other, ou can show that the diagonals share the same midpoint. Because the midpoint formula involves dividing b, if the vertices have a coefficient of, then the result will be a whole number. Houghton Mifflin Harcourt Publishing Compan Your Turn Write a coordinate proof given quadrilateral ABCD with vertices A (3, ), B (8, ), C (5, 0), and D (0, 0).. Prove that ABCD is a parallelogram. Possible answer: AB = (8-3) + ( - ) = 5; DC = AD = (3-0) + ( - 0) = 13 ; BC = 5. Prove that the diagonals of ABCD bisect each other. (5-0) + (0-0) = 5 (8-5) + ( - 0) = 13 Since the opposite sides have the same lengths, ABCD is a parallelogram. Show that the diagonals share the same midpoint. Midpoint of DB : ( 0 + 8, 0 + ) = (, 1). Midpoint of AC : (, + 0 ) = (, 1) Since the diagonals share a midpoint, the bisect each other. Module Lesson COLLABORATIVE LEARNING Small Group Activit Divide students into groups. Have each student draw a rectangle, a rhombus, and a square on a coordinate plane. Then have students trade drawings and use coordinate proofs to verif the properties of rectangles and rhombuses from the theorems the have learned Lesson 5.

4 Eplain Eample Proving Conditions for Special Parallelograms Prove or disprove each statement about the quadrilateral determined b the points Q (, -3), R (-, 0), S (-, ), and T (, 1). The diagonals of QRST are congruent. The length of SQ = The length of RT = 65. (- - ) + (0-1) = 65. ( - (-) ) + (-3 - ) = So, the diagonals of QRST are congruent. QRST is a rectangle. Find the slope of each side of the quadrilateral. Slope of QR - 1 = - = 0 - (-3) = 3-6 = - 1 ; Slope of RS = - = = = ; Slope of ST = - = = = - 1 ; Slope of TQ = - = = = Find the products of the slopes of adjacent sides. (slope of QR ) (slope of RS ) = - 1 = -1 ; (slope of RS ) (slope of ST ) = - 1 = ; -1 (slope of ST ) (slope of TQ ) = - 1 = ; (slope of TQ ) (slope of QR -1 ) = - 1 = -1 You can conclude that adjacent sides are perpendicular. So, quadrilateral QRST is a rectangle. Reflect 6. Eplain how to prove that QRST is not a square. Use the Distance Formula to compare adjacent sides; because the sides are not all congruent, QRST is not a square. R - S T 0 - Q Houghton Mifflin Harcourt Publishing Compan EXPLAIN Proving Conditions for Special Parallelograms INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Look through magazines or books to find pictures of rectangles, rhombuses, and squares. Group the pictures based on the tpe of figure. Then show the pictures to the class and ask students to describe the identifing characteristics of each tpe of figure. QUESTIONING STRATEGIES What must ou show to prove that a quadrilateral is a rectangle? How can ou show this? Show that the quadrilateral has four right angles (that is, that adjacent sides are perpendicular); show that the product of the slopes of adjacent sides is -1. Module Lesson DIFFERENTIATE INSTRUCTION Visual Cues Have students draw a rectangle, a square, and a non-rectangular parallelogram. Then have them use colored pencils to mark the diagonals as bisected, congruent, and/or perpendicular. Coordinate Proof Using Distance with Quadrilaterals 1310

5 EXPLAIN 3 Identifing Figures on the Coordinate Plane Your Turn Prove or disprove each statement about quadrilateral WXYZ determined b the points W (0, 0), X (, 3), Y (9, 3), and Z (5, 0). W Z X Y VISUAL CUES Students often confuse the relationships between diagonals of different tpes of special parallelograms. Have them review how diagonals can be used to classif rectangles, rhombuses, and squares. 7. WXYZ is a rhombus. 8. The diagonals of WXYZ are perpendicular. WX = ( - 0) + (3-0) = 5 = 5 XY = (9 - ) + (3-3) Slope of WY = = 3 = 5 = 5 YZ = (5-9) + (0-3) Slope of XZ = = -3 1 = 5 = 5 ZW = (0-5) + (0-0) (Slope of WY )(Slope of XZ ) 1 = = - 1. = 5 = 5 So, WY is perpendicular to XZ. Since all four sides have the same length, WXYZ is a rhombus. Eplain 3 Identifing Figures on the Coordinate Plane Eample 3 Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that appl. A (0, ), B (3, 6), C (8, 6), D (5, ) Houghton Mifflin Harcourt Publishing Compan Step 1 Graph ABCD. Step Determine if ABCD is a rectangle. AC = (8-0) + (6 - ) = 80 = 5 BD = (5-3) + ( - 6) = 0 = 5 Since 5 5, ABCD is not a rectangle. Thus, ABCD is not a square. Step 3 Determine if ABCD is a rhombus. Slope of AC = = 1 Slope of BD = = - Since ( 1 ) (-) = -1, AC BD. ABCD is a rhombus. 6 B C A D Module Lesson LANGUAGE SUPPORT Connect Vocabular Students have been working on the coordinate plane in both algebra and geometr. Make sure the understand what is meant b plane figures and b a coordinate plane, and how these meanings of plane are different from everda English uses of the word (such as the noun plane meaning airplane, or the verb plane, meaning to level or smooth a wood surface) Lesson 5.

6 B E(, 1), F( 3, ), G(3, 0), H(, 3) Step 1 Graph EFGH. F Step Determine if EFGH is a rectangle. EG = ( 3 - ) + ( 0 - ) = = 5 FH = ( - (-3)) + ( - ) = = 5 E Since = 5 5, the diagonals are congruent. EFGH is a rectangle. Step 3 Determine if EFGH is a rhombus. Slope of = 0 - (-1) 3 - (-) = 1 EG 7 ; Slope of FH = (- 3) = -5 5 = - 1 Since ( 1 7) (-1) -1, EG is not perpendicular to FH. So, EFGH is not a rhombus and cannot be a square. Your Turn Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that appl. 9. K( 5, 1), L(, ), M(3, 1), N(0, ) 10. P (-, 6), Q (, 5), R (3, -1), S (-3, 0) (0 - (-)) + (- - ) = 17 (3 - (-5)) + (1 - (-1)) = 17 LN = KM = Since 17 = 17, the diagonals are congruent; KLMN is a rectangle. Slope of - - LN = 0 - (-) = - 1 Slope of 1 - (-1) KM = = (-5) Since ( - 1 ) ( 1 ) = - 1, the diagonals are perpendicular; KLMN is a rhombus. Since KLMN is both a rectangle and a rhombus, it is also a square. Elaborate PR = QS = 11. How can ou use slopes to show that two line segments are parallel? Perpendicular? Parallel lines have the same slopes. Perpendicular lines have slopes that are opposite inverses, so when ou multipl them, the product will be -1. H G (3 - (-)) + (-1-6) = 7 (-3 - ) + (0-5) = 5 Since 7 5, the diagonals are not congruent. So PQRS is not a rectangle, and thus not a square. Slope of PR -1-6 = 3 - (-) = -1 Slope of QS 0-5 = -3 - = 1 Since (-1)(1) = -1, the diagonals are perpendicular, so PQRS is a rhombus. Houghton Mifflin Harcourt Publishing Compan QUESTIONING STRATEGIES How are the slopes of the sides of rectangles related? Opposite sides have equal slopes, adjacent sides have slopes with a product of -1. ELABORATE QUESTIONING STRATEGIES Can ou place a rhombus on a coordinate plane so that one side is on the -ais and one side is on the -ais? onl if the rhombus is a square SUMMARIZE THE LESSON How can ou use slope in coordinate proofs involving quadrilaterals? You can use slope to show that opposite sides of a quadrilateral have the same slope, and are therefore parallel. You can also use slope to show that the product of the slopes of an pair of adjacent sides of a quadrilateral is -1, and the sides are therefore perpendicular. 1. When ou use the distance formula, ou find the square root of a value. When finding the square root of a value, ou must consider both the positive and negative outcomes. Eplain wh the negative outcome is not used in the coordinate proofs in the lesson. Length is alwas a positive number, so the negative result of evaluating the square root does not appl to the situation. Module Lesson Coordinate Proof Using Distance with Quadrilaterals 131

7 EVALUATE 13. Essential Question Check-In How can ou use slope in coordinate proofs? You can use slope to show that opposite sides of a quadrilateral have the same slope. This shows that the opposite sides are parallel, so the quadrilateral is a parallelogram. You can also use slope to show that the product of the slopes of an pair of adjacent sides of a quadrilateral is -1. This shows that the adjacent sides are perpendicular, so the quadrilateral is a rectangle. ASSIGNMENT GUIDE Concepts and Skills Eplore Positioning a Quadrilateral on the Coordinate Plane Eample 1 Proving Properties of a Parallelogram Eample Proving Conditions for Special Parallelograms Eample 3 Identifing Figures on the Coordinate Plane Practice Eercises 1 Eercises 3 6 Eercises 7 10 Eercises Houghton Mifflin Harcourt Publishing Compan Evaluate: Homework and Practice 1. Suppose ou have a right triangle. If ou want to write a proof about the midpoints of the legs of the triangle, which placement of the triangle would be most helpful? Eplain. A. Use the origin as a verte, keeping the figure in Quadrant I with vertices (0, b), (a, 0), and (0, 0). B. Center the triangle at the origin. C. Use the origin as a verte, keeping the figure in Quadrant I with vertices (0, b), (a, 0), and (0, 0). D. Center one leg of the triangle on the -ais with vertices (0, a), (0, a), and (b, a). E. Use the -ais as one leg of the triangle with vertices (a, 0), (a, b), and (a + c, 0). A; Use the origin as a verte, keeping the figure in Quadrant I with vertices (0, b), (a, 0), and (0, 0). Positioning the triangle in Quadrant I will mean that all the vertices are positive. Also, if ou need to find the midpoints of the sides, using a coefficient of for the vertices will mean that the coordinates of the midpoint will be whole numbers.. Describe the position of a general isosceles trapezoid WXYZ determined b the points W (0, 0), X (a, 0), Y (a c, b), and Z (c, b). Then sketch the trapezoid. The origin is a verte, and the -ais contains one side of the trapezoid. Write a coordinate proof for the quadrilateral determined b the points A (, ), B (, 1), C ( 1, 3), and D ( 3, ). W (0, 0) Z (c, b) Online Homework Hints and Help Etra Practice Y (a - c, b) X (a, 0) 3. Prove that ABCD is a parallelogram.. Prove that ABCD is a rectangle. Slope of AB = -5 ; Slope of BC = 5 ; Since we alread proved that ABCD is a Slope of -5 DC ; Slope of DA = parallelogram, if the diagonals are 5 The slopes of opposite sides are equal. congruent then ABCD is a rectangle. This means that opposite sides are AC = 58 ; DB = 58 parallel, so ABCD is a parallelogram. The diagonals are congruent, so ABCD is a rectangle. Module Lesson 1313 Lesson 5. Eercise Depth of Knowledge (D.O.K.) Mathematical Practices 1 10 Skills/Concepts MP.3 Logic Strategic Thinking MP. Reasoning 17 1 Skills/Concepts MP. Modeling Skills/Concepts MP.1 Problem Solving 3 3 Strategic Thinking MP. Reasoning 3 Strategic Thinking MP. Reasoning 5 3 Strategic Thinking MP. Reasoning 6 3 Strategic Thinking MP. Reasoning

8 5. Prove that ABCD is a rhombus. 6. Prove that ABCD is a square. In Evaluate3 we proved that ABCD is a parallelogram. Slope of 7 AC = 3 ; Slope of DB = -3 7 Since ( 7 3 )( -3 7 ) = -1, the diagonals are perpendicular. So, ABCD is a rhombus. Since ABCD is both a rectangle and rhombus, it is also a square. Prove or disprove each statement about the quadrilateral determined b the points W (, 5), X (5, 5), Y (5, 0), and Z (, 0). VISUAL CUES Some students ma have difficult using the formula to calculate slopes. You might suggest that these students tr a more visual approach. Have them place the figure on a coordinate plane, then determine the rise and run visuall. 7. Prove that the diagonals are congruent. 8. Prove that the diagonals are perpendicular. WY = (- - 5) + (5-0) Slope of WY = -5 = 7 XZ = (- - 5) + (0-5) 7 ; Slope of 5 XZ = 7 When ou multipl the slopes, the product is = 7 not -1: So the diagonals are congruent. ( -5 7 )( 5 7 ) = -5. So the diagonals 9 are not perpendicular. 9. Prove that the diagonals bisect each other. 10. Prove that WXYZ is a square. midpoint of WY = (, ) = ( 3 5, Since the diagonals bisect each other, WXYZ ) is a parallelogram. Since the diagonals are midpoint of XZ = ( 5 + (-) 5 + 0, ) = ( 3 5 congruent, WXYZ is a rectangle. Since the, ) diagonals are not perpendicular, it is not a Since the share a midpoint, the rhombus. So, it is not a square. diagonals bisect each other. Algebra Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that appl. 11. A ( 10, ), B (, 10), C (, ), D (, ) 1. J( 9, 7), K(, ), L(3, 3), M(, 8) Slope of BD = 7; Slope of AC = Since (7) ( Slope of JL = -1 3 ; Slope of KM = -3 7 ) = -1, the diagonals are Since ( 1 3 )(-3) = -1, the diagonals are perpendicular. ABCD is a rhombus. perpendicular. JKLM is a rhombus. BD = AC = 00 JL = 160 ; KM = 0 Since the lengths of the diagonals are equal. ABCD is a rectangle. Since ABCD is both a rhombus and a rectangle, it is also a square. Since the diagonals are not congruent, JKLM is not a rectangle. So JKLM is not a square either. Houghton Mifflin Harcourt Publishing Compan Module Lesson Coordinate Proof Using Distance with Quadrilaterals 131

9 QUESTIONING STRATEGIES How do ou find the lengths of the diagonals of a quadrilateral in the coordinate plane? Use the distance formula to find the distance between the coordinates of the endpoints of each diagonal. Analze Relationships The coordinates of three vertices of parallelogram ABCD are given. Find the coordinates of the fourth point so that the given tpe of figure is formed. 13. A (, ), B ( 5, ), D (, ), rectangle 1. A ( 5, 5), B (0, 0), C (7, 1), rhombus Slope of AB = 0-9 = 0 For ABCD to be a rhombus, opposite sides should be Slope of 6 parallel and all four sides should be congruent. AD = so the slope is undefined. 0 Slope from B to A: AD is a vertical line. For ABCD to be a rectangle, rise = 5-0 = 5 BC must be parallel to AD. Thus, the slope of run = -5-0 = -5 BC must be undefined. So, Use the same slope from C to D: Start at (7, 1) and BC must be a vertical line and move 5 units to the left and 5 units up. D is at (, 6). C must be (-5, ). Slope of AB = -1; Slope of 1 BC = 7 ; Slope of CD = = = -1; Slope of DA = -5 - = = 7 Opposite sides are parallel. Check: AB = BC = CD = AD = 5 ABCD is a rhombus. Houghton Mifflin Harcourt Publishing Compan 15. A (0, ), B (, ), C (0, 6), square 16. A (, 1), B ( 1, 5), C ( 5, ), square Show that the product of the slopes of Show that the product of the slopes of adjacent adjacent sides is -1 and that all four sides sides is -1 and that all four sides are congruent. are congruent. Slope from B to A: Slope from B to A: rise = 1-5 = - rise = - (-) = run = - (-1) = 3 run = 0 - = - Use the same slope from C to D: Use the same slope from C to D: Start at Start at (-5, ) and move down units and (0, -6) and move 3 units to the right. D is at (-, -). units to the left and units up. D is at Slope of AB = - (-, -). Slope of AB = -1; Slope of 3 ; Slope of 3 BC = ; BC = 1; Slope of CD = -1; Slope of Slope of CD = - 3 ; Slope of 3 DA = DA = 1 Adjacent sides are perpendicular. Adjacent sides are perpendicular. Check: AB = BC = CD = AD = 5 Check: AB = BC = CD = AD = ABCD is a square. ABCD is a square. Paul designed a doghouse to fit against the side of his house. His plan consisted of a right triangle on top of a rectangle. Use the drawing for Eercises Find BD, CE, and BE. BD = BC + CD = AE + CD = = 38 in. DE = C D + C E ; CE = D E - C D ; CE = 6-10 = B E = B C + E C ; BE = B C + E C = 8 + = 576 = in in. E 8 in. A D 6 in. 10 in. C B 18. Before building the doghouse, Paul sketched his plan on a coordinate plane. He placed A at the origin and AB on the -ais. Find the coordinates of B, C, D, and E, assuming that each unit of the coordinate plane represents one inch. B = (, 0) ; C = (, 8) ; D = (, 38) ; E = (0, 8) Module Lesson 1315 Lesson 5.

10 19. Critical Thinking On the National Mall in Washington, D.C., a reflecting pool lies between the Lincoln Memorial and the World War II Memorial. The pool has two 300-foot-long sides and two 150-foot-long sides. Tell what additional information ou need to know in order to determine whether the reflecting pool is a rectangle. (Hint: Remember that ou have to show it is a parallelogram first.) Possible answer: To know that the reflecting pool is a parallelogram, the congruent sides must be opposite each other. If this is true, then knowing that one angle in the pool is a right angle or that the diagonals are congruent proves that the pool is a rectangle. AVOID COMMON ERRORS Students ma unintentionall make an eercise harder than it needs to be b positioning a figure over positive and negative quadrants. Advise them to keep their figures in one quadrant if possible, preferabl Quadrant I. Algebra Write a coordinate proof. 0. The Bushmen in South Africa use the Global Positioning Sstem to transmit data about endangered animals to conservationists. The Bushmen have sighted animals at the following coordinates: ( 5, 31.5), ( 3., 31.), and (, 31.1). Prove that the distance between two of these locations is approimatel twice the distance between two other locations. (-3. - (-5)) + ( ) 1.8 units (- - (-3.)) + ( ) 0.9 units (- - (-5)) + ( ) 1.1 units 1.8 is twice 0.9. The distance between of the locations is appro. twice the distance between another locations. 1. Two cruise ships leave a port located at P (10, 50). One ship sails to an island located at A ( 0, 10), and the other sails to an island located at B (60, 110). Suppose that each unit represents one nautical mile. Find the midpoint of the line segment connecting the two cruise ships. Verif that the port and the two cruise ships are in a line. Midpoint of AB = (, ) = (10, 50) Since P is the midpoint of AB it lies on AB. Therefore, points A, B, and P are collinear and the port and the two cruise ships are in a line. Houghton Mifflin Harcourt Publishing Compan Image Credits: (t) Image Source/Alam; (c) Stefanie van der Vin/Fotolia; (b) icholakov/istockphoto.com Module Lesson Coordinate Proof Using Distance with Quadrilaterals 1316

11 JOURNAL Have students write a journal entr in which the give the coordinates of four points and then prove or disprove that the quadrilateral determined b the points is a parallelogram or rectangle.. A parallelogram has vertices at (0, 0), (5, 6), and (10, 0). Which could be the fourth verte of the parallelogram? Choose all that appl. A. (5, -6) B. (15, 6) C. (0, -6) D. (10, 6) A, B, D, E; these coordinates result in a parallelogram because opposite sides have equal slopes. E. (-5, 6) H.O.T. Focus on Higher Order Thinking 3. Draw Conclusions The diagonals of a parallelogram intersect at (-, 1.5). Two vertices are located at (-7, ) and (, 6.5). Find the coordinates of the other two vertices. Let A be at (-7, ), B at (, 6.5), C at (, ), D at (a, b), and E at (-, 1.5) Midpoint of AC = ( -7 +, + ) = (-, 1.5), so -7 + = - and + = 1.5. = 3 and = 1, so point C is at (3.1). Midpoint of BD = ( + a, b ) = (-, 1.5), so + a a = -6 and b = -3.5, so point D is at (-6, -3.5). Analze Relationships Consider points L (3, -), M (1, -), and N (5, ). = - and b = 1.5. a. Find coordinates for point P so that the quadrilateral determined b points L, M, N, and P is a parallelogram. Is there more than one possibilit? Eplain. The coordinates of P ma be (7, 0), (3, ), or (-1, -8). These coordinates result in a parallelogram because opposite sides have equal slopes. Houghton Mifflin Harcourt Publishing Compan b. Are an of the parallelograms a rectangle? Wh? When the coordinates of P are (7, 0), the quadrilateral is a rectangle because the product of the slopes of adjacent sides is Critical Thinking Rhombus OPQR has vertices O (0, 0), P (a, b), Q (a + b, a + b), and R (b, a). Prove the diagonals of the rhombus are perpendicular. Slope of (a + b) - 0 OQ = (a + b) - 0 = a + b a + b = 1; Slope of PR = a - b b - a = a - b -1(a - b) = -1 Because the product of the slopes is -1, the diagonals are perpendicular. Module Lesson 1317 Lesson 5.

12 6. Multi-Step Use coordinates to verif the Trapezoid Midsegment Theorem which states The midsegment of a trapezoid is parallel to each base, and its length is one half the sum of the lengths of the bases. a. M is the midpoint of QP. What are its coordinates? ( a + 0, b + 0 ) = (a, b) b. N is the midpoint of RS. What are its coordinates? c + c + d (, b ) ( = (c + d) b, ) = (c + d, b) c. Find the slopes of QR, PS, MN. What can ou conclude? Slope of b - b QR = 0 c - a = c - a = 0; Slope of PS = c + d - 0 = c + d = 0; Slope of MN = b - b 0 c + d - a = c + d - a = 0 All three d. Find QR, PS line, MN segments are parallel, and are horizontal lines.. Show that MN = 1 (PS + QR). QR = (c - a) + (b - b) = (c - a) = c - a PS = (c + d - 0) + (0-0) = MN = (c + d - a) + (b - b) = 1 Lesson Performance Task According to the new maor, the shape of Cit Park is downright ugl. While the parks in all of the other towns in the vicinit have nice, regular polgonal shapes, Cit Park is the shape of an irregular quadrilateral. On a coordinate map of the park, the four corners are located at (-3, ), (5, ), (1, -), and (-5, -). The maor s chief assistant knows a little mathematics and proposes that a special inner park be created b joining the midpoints of the sides of Cit Park. The assistant claims that the boundaries of the inner park will create a nice, regular polgonal shape, just like the parks in all the other towns. The maor thinks the idea is ridiculous, saing, You can t create order out of chaos. 1. Who was right? Eplain our reasoning in detail.. Irregular quadrilateral ABCD is shown here. Points J, K, L, and M are midpoints. a. What must ou show to prove that quadrilateral JKLM is a parallelogram? b. How can ou show this? (c + d) = c + d c. If the adjacent sides of JKLM are perpendicular, what tpe of figure does that make JKLM? 1. The assistant was right. The coordinates of the inner park are (1, 3), (3, 0), (-, -3), A J M Q (a, b) R (c, b) M N P (0, 0) S (c + d, 0) (c + d - a) = c + d - a (PS + QR) = 1 (c + d + c - a) = 1 (c + d - a) = 1 ( (c + d -a)) = c + d - a = MN and (-, 0). The slopes of opposite sides are equal. One pair of sides has a slope of 3 5. The other pair of sides has a slope of - 3. This means that inner park is a parallelogram.. a. Show that both pairs of opposite sides are parallel. b. Show that slopes of both pairs of opposite sides are equal. c. If adjacent sides are perpendicular, then quadrilateral JKLM would be a rectangle. B D K L C Houghton Mifflin Harcourt Publishing Compan AVOID COMMON ERRORS The Distance Formula contains the terms ( - 1 ) and ( - 1 ) The Midpoint Formula contains the terms ( 1 + ) and ( 1 + ). Because the terms are so similar, the are easil confused, particularl when students are just beginning to use the formulas. Caution students to be sure the re using the Midpoint Formula correctl when the find the coordinates of the midpoints of the quadrilateral in the Lesson Performance Task. INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP. In the Lesson Performance Task, quadrilateral JKLM is shown to be a parallelogram b showing that its opposite sides JK and LM are both congruent and parallel. Eplain how ou could prove that JKLM is a parallelogram b showing that both pairs of opposite sides are congruent. Sample answer: JK LM as shown in the Lesson Performance Task. Draw auiliar line segment BD. B the Midline Theorem, both JM and KL are half the length of BD. Therefore, JM KL. Since JK LM and JM KL, JKLM is a parallelogram. Module Lesson EXTENSION ACTIVITY In the Lesson Performance Activit, students use the Midline Theorem to prove that the figure formed b joining the midpoints of a quadrilateral an quadrilateral is a parallelogram. Even after this is proven, the conclusion remains one of the most amazing in all of plane geometr. (It holds even for concave quadrilaterals and quadrilaterals with intersecting sides.) Have students draw a collection of quadrilaterals of widel varing shapes and side lengths. Urge them to draw the most irregular-looking quadrilaterals the can think of. For each, students should measure the sides to find the midpoints and then join them to create a quadrilateral. The can then measure the angles or side lengths of the new quadrilateral to show that it is a parallelogram. Scoring Rubric points: Student correctl solves the problem and eplains his/her reasoning. 1 point: Student shows good understanding of the problem but does not full solve or eplain. 0 points: Student does not demonstrate understanding of the problem. Coordinate Proof Using Distance with Quadrilaterals 1318