Proofs Using Coordinate Geometry

1. Proofs Using Coordinate Geometry Lesson 6-7 Check Skills You’ll Need (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. rectangle 3. isosceles triangle Check Skills You’ll Need 6-7

2. Proofs Using Coordinate Geometry Lesson 6-7 Check Skills You’ll Need Solutions 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. 1 2 3 2 9 2 11 2 rise run 2 4 1 2 –2 1 1 2 6-7

3. 2. The coordinates of B are (a, c). 3. Since BOAO the coordinates of B are (–a, 0). Proofs Using Coordinate Geometry Lesson 6-7 Check Skills You’ll Need Solutions (continued) 6-7

4. b a – b 2 c + a 2 b 2 a 2 midpoint: , ; slope: midpoint: , ; slope: c + a b Placing Figures in the Coordinate Plane Lesson 6-6 Lesson Quiz Find the missing coordinates of each figure. 2. rhombus 3. rectangle 1. parallelogram M (b, c + a) M(2a, 0), D(a, –b) A(0, b), D(a, 0) Find the coordinates of the midpoint and the slope. 4.OM in Exercise 1 5. AD in Exercise 2 6. AD in Exercise 3 midpoint: (a, 0); slope: undefined 6-7

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

7. Placing Figures in the Coordinate Plane Lesson 6-7 Notes 6-7

8. 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 Additional Examples 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. Quick Check 6-7

9. The quadrilateral XYZW formed by connecting the midpoints of ABCD is shown below. Proofs Using Coordinate Geometry Lesson 6-7 Additional Examples Use coordinate geometry to prove that the quadrilateral formed by connecting the midpoints of rhombus ABCD is a rectangle. 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. 6-7

10. ( –2a)2 + (2b)2 = 4a2 + 4b2 XZ = (–a – a)2 + (b – (–b))2 = ( –2a)2 + (–2b)2 = 4a2 + 4b2 YW = (–a – a)2 + (– b – b)2 = Proofs Using Coordinate Geometry Lesson 6-7 Additional Examples Quick Check (continued) To show that XYZW is a rectangle, find the lengths of its diagonals, and then compare them to show that they are equal. Because the diagonals are congruent, parallelogram XYZW is a rectangle. 6-7

11. Proofs Using Coordinate Geometry Lesson 6-7 Lesson Quiz 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. (a + c, d ) (b + c, d ) 3. Explain how you know that MN || AB. 4. Show that MN = AB. 5. What theorem do Exercises 1–4 prove? Both have slope 0, so they are parallel. 1 2 The Distance Formula finds MN = b – a and AB = 2b – 2a. Triangle Midsegment Theorem 6-7