In triangle XYZ, XY is 13, YZ is 15 and ZX is 14.

[BTGModeratorVI]

In triangle XYZ, XY is 13, YZ is 15 and ZX is 14. What is the height of the triangle in inches as measured from point Y to XZ?

A. 4
B. 6
C. 8
D. 10
E. 12

Source: Target Test Prep
Answer: E

[Brent@GMATPrepNow]

In triangle XYZ, XY is 13, YZ is 15 and ZX is 14. What is the height of the triangle in inches as measured from point Y to XZ?

A. 4
B. 6
C. 8
D. 10
E. 12

Source: Target Test Prep
Answer: E

Given:

Our goal is to find the value of h

ASIDE: Since all of the answer choices are integers, we COULD test some Pythagorean triplets (3-4-5 or 5-12-13 or 9-12-15 etc) until we find that h = 12 (and k = 5)
However, if we don't spot that shortcut we can always use some algebra.

First notice that we can divide XZ into two lengths: k and 14 - k


At this point, we can see that we have two different right triangles.


So, when we apply the Pythagorean theorem we get two equations:
h² + (14-k)² = 15²
h² + k² = 13²

Expand and simplify the top equation to get:
h² + 196 - 28k + k² = 225
h² + k² = 169

Subtract at the bottom equation from the top equation to get: 196 - 28k = 56
Solve: k = 5

Now that we know k = 5, we can see that the GREEN right triangle must be a 5-12-13 right triangle, which means h = 12

Answer: E

Cheers,
Brent