AMSCO'S Geometry. New York - Rye High School

522 Ratio, Proportion, and SimilarityLet B(x 1, y 1) and A(x 2, y 2) be any twopoints in the coordinate plane. From Adraw a vertical line and from B draw ahorizontal line. Let the intersection ofthese two lines be C. The coordinates ofC are (x 2, y 1). Let AB c, CB a x 2 x 1, and CA b y 2 y 1. Then,yOA(x 2, y 2)cbaB(x 1, y 1) C(x 2, y 1)xc 2 a 2 b 2c 2 x 2 x 1 2 y 2 y 1 2c "(x 22 x 1) 2 1 (y 22 y 1) 2This result is called the distance formula. If the endpoints of a line segmentin the coordinate plane are B(x 1, y 1) and A(x 2, y 2), then:AB "(x 22 x 1) 2 1 (y 22 y 1) 2EXAMPLE 1The coordinates of the vertices of quadrilateralABCD are A(1, 3), B(6, 4), C(5, 3), andD(2, 4).a. Prove that ABCD is a rhombus.DyCb. Prove that ABCD is not a square.11OxSolutionABa.AB 5 "(6 2 (21)) 2 1 (24 2 (23)) 2 BC 5 "(5 2 6) 2 1 (3 2 (24)) 25 "(7) 2 1 (21) 2 5 "(21) 2 1 (7) 25 "49 1 15 "505 "50CD 5 "(22 2 5) 2 1 (4 2 3) 2DA 5 "(21 2 (22)) 2 1 (23 2 4) 25 "(27) 2 1 (1) 2 5 "(1) 2 1 (27) 25 "49 1 15 "505 "1 1 495 "505 "1 1 49