Triangle $abc$ has side lengths $ab = 9$, $ac = 10$, and $bc = 17$. let $x$ be the intersection of the angle bisector of $\angle a$ with side $\overline{bc}$, and let $y$ be the foot of the perpendicular from $x$ to side $\overline{ac}$. compute the length of $\overline{xy}$.

Step-by-step explanation:

Consider triangle ABC. Segment AX is angle A bisector. Its length can be calculated using formula

Hence,

By the angle bisector theorem,

So,

and

By the Pythagorean theorem for the right triangles AXY and CXY:

Thus,

Subtract from the second equation the first one:

Hence,

Step-by-step explanation:

The semiperimeter of triangle is , so by Heron's formula, the area of triangle is .

The area of triangle $PRX$ is

.By the angle bisector theorem, we have , so and , which means

.

Then is the height of triangle with respect to base , so , which gives us

.

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