+13 Right triangle ABC has side lenghts AB = 3, BC = 4, and AC = 5. Square XYZW is inscribed in triangle ABC with X and Y on line AB, and Z on
line BC. What is the side length of the square
+1639 This is an interesting answer:
Using similar triangles twice, and calling the length of the side of the square x,
in the triangle CYZ, CY/x = 4/3, so CY = 4x/3,
in the triangle WXA, x/XA = 3/4, so XA = 3x/4.
The hypotenuse of the triangle is
CY + YX + XA = 4x/3 + x + 3x/4 = 37x/12 = 5,
so x = 60/37.
(Solved by Guest - Nov 13, 2015)
+128475 Call the side of the square = s
Triangle ABC is similar to triangle AXW
So
AB/BC = AX /XW
3/ 4 = AX / s
AX = (3/4)s
Triangle ABC is also similar to triangle ZYC
So
AB/ BC = ZY/YC
3/4 = s/ YC
YC = (4/3)s
And
AX + XY + YC = 5
(3/4)s + s + (4/3)s = 5
(9/12)s + (12/12) + (16/12)s = 5
(37/12)s = 5
s = 5(12/37) = 60/37
