Given right triangle A w/a hypotenuse length of 3y + x and a leg of y-x, and right triangle B, w / a hypotense length of y+5 and a leg length of x+5, for what values of x and y are the triangles congruent by HL?

Triangle A

hypotenuse: 3y + x

leg: y - x

Triangle B

hypotenuse: y + 5

leg: x + 5

Congruency = > 3y + x = y + 5 and y - x = x + 5

Solve the system

3y + x = y + 5 - > 2y + x = 5

y - x = x + 5 - > y - 2x = 5

2y + x = 5

-2y + 4x = - 10

5x = - 5

x = - 1

y = (5 - (-1)) / 2 = 6/2 = 3

Verify:

hypotenuses

3y + x = 9 - 1 = 8

y + 5 = 3 + 5 = 8

Legs:

y - x = 3 - (-1) = 3 + 1 = 4

x + 5 = - 1 + 5 = 4.

Then both hypotenuses and both legs are congruent.

Answer: x = - 1 and y = 3