Problem:
Right △ABC has AB=3,BC=4, and AC=5. Square XYZW is inscribed in △ABC with X and Y on AC,W on AB, and Z on BC. What is the side length of the square?
Answer Choices:
A. 23
B. 3760
C. 712
D. 1323
E. 2
Solution:
Let s be the side length of the square, and let h be the length of the altitude of △ABC from B. Because △ABC and △WBZ are similar, it follows that
sh−s=ACh=5h, so s=5+h5h
Because h=3⋅4/5=12/5, the side length of the square is
s=5+12/55(12/5)=3760
OR
Because △WBZ is similar to △ABC, we have
BZ=54s and CZ=4−54s
Because △ZYC is similar to △ABC, we have
4−(4/5)ss=53
Thus
5s=12−512s and s=(B)3760
The problems on this page are the property of the MAA's American Mathematics Competitions
