In isosceles right triangle ABC, shown here, AC = BC. Point X is on side BC such that CX = 6 and XB = 12, and Y is on side AB such that XY is perpendicular to AB. What is the ratio of BY to YA?
+129845 Note that triangles BYX and BCA are similar by AA congruency
And AC = BC so XY = BY
And since BX = 12......then BX = sqrt ( XY^2 + BY^2) = sqrt (2*BY^2) = BY * sqrt 2
So BY = 12/sqrt 2 = 6sqrt 2
Likewise....since AC = BC then
BA = sqrt ( AC^2 + BC^2) = sqrt [ 2 * AC^2] = AC * sqrt 2 = 18sqrt 2
So YA = BA -BY = 18sqrt 2 - 6sqrt 2 = 12 sqrt 2
So
BY / YA = [ 6sqrt 2 ] / [12 sqrt 2 ] = 6/12 = 1/2
