AMSCO'S Geometry. New York - Rye High School

302 Slopes and Equations of LinesNow draw a vertical line through Sand a horizontal line through T. Theselines appear to intersect at a point onPQ that we will call M. This point hasthe coordinates (5, 3). We need to showthat this point is a point on PQ and is themidpoint of PQ.The point M is on PQ if and onlyif the slope of PM is equal to the slopeof MQ.y1O1M(5, 3)P(2, 1) S(5, 1)Q(8, 5)T(8, 3)R(8, 1)xslope ofPM 5 3 2 15 2 2slope ofMQ 5 5 2 38 2 55 2 35 2 3Since these slopes are equal, P, M, and Q lie on a line.The point M is the midpoint of PQ if PM MQ. We can show thatPM MQ by showing that they are corresponding parts of congruent triangles.• PS 5 2 3 and MT 8 5 3. yTherefore, PS > MT.Q(8, 5)• SM 3 1 2 and TQ 5 3 2.M(5, 3)T(8, 3)Therefore, SM > TQ.• Since vertical lines are perpendicular 1R(8, 1)to horizontal lines, PSM andO P(2, 1) S(5, 1)MTQ are right angles and thereforecongruent.1x• Therefore, PSM MTQ by SAS and PM > MQ because correspondingparts of congruent triangles are congruent.We can conclude that the coordinates of the midpoint of a line segmentwhose endpoints are (2, 1) and (8, 5) are A 2 1 2 8 , 1 1 2 5 B (5, 3).This example suggests the following theorem:Theorem 8.1If the endpoints of a line segment are (x 1, y 1) and (x 2, y 2), then the coordinatesof the midpoint of the segment are A x 1 1 x 22 , y 1 1 y 22 B .Given The endpoints of AB are A(x 1, y 1) andB(x 2, y 2).Prove The coordinates of the midpoint of ABare A x 1 1 x 22 , y 1 1 y 22 B .yOB(x 2, y 2)xM( 1 x 2 y, 1 y 2)A(x 1, y 1)x22