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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 Nov 21, 2020

#1
+4

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)

Nov 21, 2020
#3
+5

That's nice, but I like my method too. Let's denote a side of a square with a letter  a

Tangent of angle BAC = 1.33333333

Tangent of angle BCA = 0.75

So, we have     (a / 1.3333333) + a + (a / 0.75) = 5  ==>   a = 1.621621622     or     60 / 37

jugoslav  Nov 21, 2020
edited by jugoslav  Nov 21, 2020
#2
+4

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   Nov 21, 2020