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FeatureLesson Geometry Lesson Main (For help, go to Lesson 6-6.) 1.Graph the rhombus with vertices A(2, 2), B(7, 2), C(4, –2), and D(–1, –2). Then, connect the midpoints of consecutive sides to form a quadrilateral. What do you notice about the quadrilateral? Give the coordinates of B without using any new variables. 2.rectangle3.isosceles triangle Proofs Using Coordinate Geometry Lesson 6-7 Check Skills Youll Need 6-7

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FeatureLesson Geometry Lesson Main 1.The coordinates of the midpoints are (, 2 ), (, 0 ), (, –2 ), and (, 0 ). The slope ( ) of the two longer segments is, or, and the slope of the two shorter segments is, or –2. Since the slopes of opposite sides are equal, opposite sides are parallel. The quadrilateral is a parallelogram. Also, since (–2) = –1, adjacent sides are perpendicular to one another. Thus, the quadrilateral is a rectangle. rise run 9292 11 2 3232 1212 2424 1212 –2 1 1212 Solutions Proofs Using Coordinate Geometry Lesson 6-7 Check Skills Youll Need 6-7

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FeatureLesson Geometry Lesson Main 2.The coordinates of B are (a, c). 3.Since BO AO the coordinates of B are (–a, 0). Solutions (continued) Proofs Using Coordinate Geometry Lesson 6-7 Check Skills Youll Need 6-7

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FeatureLesson Geometry Lesson Main Find the missing coordinates of each figure. 1. parallelogram 2. rhombus3. rectangle M (b, c + a) M(2a, 0), D(a, –b) A(0, b), D(a, 0) Placing Figures in the Coordinate Plane Lesson 6-6 Find the coordinates of the midpoint and the slope. 4.OM in Exercise 15. AD in Exercise 26. AD in Exercise 3 midpoint:, ; slope: b2b2 c + a 2 c + a b midpoint: (a, 0); slope: undefined midpoint:, ; slope: a2a2 b2b2 baba – Lesson Quiz 6-7

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FeatureLesson Geometry Lesson Main

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FeatureLesson Geometry Lesson Main Placing Figures in the Coordinate Plane Lesson 6-7 Notes 6-7 The midsegment of a trapezoid is the segment that joins the midpoints of the nonparallel opposite sides.

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FeatureLesson Geometry Lesson Main Placing Figures in the Coordinate Plane Lesson 6-7 Notes 6-7

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FeatureLesson Geometry Lesson Main Examine trapezoid TRAP. Explain why you can assign the same y-coordinate to points R and A. The y-coordinates of all points on a horizontal line are the same, so points R and A have the same y-coordinates. In a trapezoid, only one pair of sides is parallel. In TRAP, TP || RA. Because TP lies on the horizontal x-axis, RA also must be horizontal. Proofs Using Coordinate Geometry Lesson 6-7 Quick Check Additional Examples 6-7

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FeatureLesson Geometry Lesson Main Use coordinate geometry to prove that the quadrilateral formed by connecting the midpoints of rhombus ABCD is a rectangle. Proofs Using Coordinate Geometry Lesson 6-7 The quadrilateral XYZW formed by connecting the midpoints of ABCD is shown below. From Lesson 6-6, you know that XYZW is a parallelogram. If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle by Theorem 6-14. Additional Examples 6-7

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FeatureLesson Geometry Lesson Main Because the diagonals are congruent, parallelogram XYZW is a rectangle. To show that XYZW is a rectangle, find the lengths of its diagonals, and then compare them to show that they are equal. Proofs Using Coordinate Geometry Lesson 6-7 (continued) Quick Check XZ = (–a – a) 2 + (b – (–b)) 2 =( –2a) 2 + (2b) 2 = 4a 2 + 4b 2 YW = (–a – a) 2 + (– b – b) 2 = ( –2a) 2 + (–2b) 2 = 4a 2 + 4b 2 Additional Examples 6-7

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FeatureLesson Geometry Lesson Main Use the diagram for Exercises 1–5. 1. Point M is the midpoint of AC. Find its coordinates. 2. Point N is the midpoint of BC. Find its coordinates. 3.Explain how you know that MN || AB. 4.Show that MN = AB. 5.What theorem do Exercises 1–4 prove? Triangle Midsegment Theorem (b + c, d) 1212 Both have slope 0, so they are parallel. The Distance Formula finds MN = b – a and AB = 2b – 2a. (a + c, d) Proofs Using Coordinate Geometry Lesson 6-7 Lesson Quiz 6-7

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