Question

the sides of a right angle triangle are in arithmetic progression show that the sides are in the ratio 3 is to 4 is to 5

Answer 1

down voteLet's call the first term aa and common difference dd. Therefore, your sequence becomes:a, a+d, a+2da, a+d, a+2dNow, these 3 terms form the sides of a right triangle. Since a+2da+2d is the largest (assuming d > 0), we can say that it is the hypothenuse.Using the Pythagorean Theorem:a2+(a+d)2=(a+2d)2a2+(a+d)2=(a+2d)2a2+a2+2ad+d2=a2+4ad+4d2a2+a2+2ad+d2=a2+4ad+4d23d2−a2+2ad=03d2−a2+2ad=0(3d−a)(d+a)=0(3d−a)(d+a)=0Therefore, your solutions are3d=a3d=aandd=−ad=−aSo if a = 3, d = 1. Therfore, one possible sequence is 3, 4, 5.Similarly, if a = 6, then d = 2, so another possible sequence is 6, 8, and 10.

Hope This Helps :)