AMSCO'S Geometry. New York - Rye High School

Midpoint of a Line Segment 303ProofIn this proof we will use the following facts from previous chapters that wehave shown to be true:• Three points lie on the same line if the slope of the segment joining two ofthe points is equal to the slope of the segment joining one of these pointsto the third.• If two points lie on the same horizontal line, they have the same y-coordinateand the length of the segment joining them is the absolute value ofthe difference of their x-coordinates.• If two points lie on the same vertical line, they have the same x-coordinateand the length of the segment joining them is the absolute value of the differenceof their y-coordinates.We will follow a strategy similar to the one used in the previous example. First,we will prove that the point M with coordinates A x 1 1 x 22 , y 1 1 y 22 B is on AB, andthen we will use congruent triangles to show that AM > MB. From the definitionof a midpoint of a segment, this will prove that M is the midpoint of AB.(1) Show that M A x 1 1 x 22 , y 1 1 y 22 B lies on AB:y 11 y 22 2 y 1 y 22 y 1 1 y 2slope of AM 5 x slope of MB 511 x 22 2 x 1x 22 x 1 1 2x 225 2y 2 2 (y 1 1 y 25 y 1 1 y 2 2 2y 1)x 1 1 x 2 2 2x 1 2x 2 2 (x 1 1 x 2 )5 y 2 2 y 15 y 2 2 y 1x 2 2 x 1 x 2 2 x 1Points A, M, and B lie on the same line because the slope of AM is equalto the slope of MB.(2) Let C be the point on the same vertical line as B and the same horizontalline as A. The coordinates of C are (x 2, y 1).yA(x 1, y 1)The midpoint of is D A x 1 1 x 2AC 2 , y 1 B .The midpoint of is E A x 2, y 1 1 y 2BC2 B .O2x 1 x 2y 1 y 22M( , )xD( 1 x 22 , y 1)B(x 2, y 2)yE(x 2, 1 y 22 )C(x 2, y 1)x