Show that TWVU is a parallelogram by proving both pairs of opposite sides congruent.

1. Placing Figures in the Coordinate Plane LESSON 6-6 Additional Examples Show that TWVU is a parallelogram by proving both pairs of opposite sides congruent. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram by Theorem 6-6. You can prove that TWVU is a parallelogram by showing that TW = VU and WV = TU. Use the Distance Formula.

2. TW = (a + c – a)2 + (b + d – b)2 = VU = (c + e – e)2 + (d – 0)2 = WV = ((a + c) – (c + e))2 + (b + d – d)2 = TU = (a – e)2 + (b – 0)2 = (a – e)2 + b2 (a – e)2 + b2 c2 + d2 c2 + d2 Placing Figures in the Coordinate Plane LESSON 6-6 Additional Examples (continued) Use the coordinates T(a, b), W(a + c, b + d), V(c + e, d), and U(e, 0) that you found in Example 2. Because TW = VU and WV = TU, TWVU is a parallelogram. Quick Check

3. Because AB || CO and CO is horizontal, AB is also horizontal. So point B has the same second coordinate, q, as point A. Placing Figures in the Coordinate Plane LESSON 6-6 Additional Examples Use the properties of parallelogram OCBA to find the missing coordinates. Do not use any new variables. The vertex O is the origin with coordinates O(0, 0). Because point A is p units to the left of point O, point B is also p units to the left of point C because OCBA is a parallelogram. So the first coordinate of point B is –p – x. The missing coordinates are O(0, 0) and B(–p – x, q). Quick Check