Triangle ABC is a right triangle with ∠ABC=90∘. P is a point within triangle ABC such that ∠APB=∠BPC=∠CPA=120∘. If PA=15 and PB=6, what is the value of PC?

There's probably a good geometric way to do it, but here's a trig way, using the law of cosines.

AB^2 = 6^2+15^2 - 2 * 6 * 15 * (-1/2) = 351

AC^2 = PC^2 + 15^2 - 2*PC*15(-1/2)
BC^2 = PC^2 + 6^2 - 2*PC*6(-1/2)

AC^2-BC^2 = AB^2 = 351, so

351 = 189 + 9PC
9PC = 162

To find the value of PC in triangle ABC, we can use the Law of Sines. The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant:

a/sin(A) = b/sin(B) = c/sin(C)

In triangle ABC, we are given the lengths of two sides and the value of an angle. Let's label the length of PC as c, and the measure of angle ACB as C.

We know that PA = 15 and PB = 6. Let's label the length of AB as a. Since angle ACB is a right angle, we can use the Pythagorean theorem to find the length of AB:

AB^2 = PA^2 + PB^2
AB^2 = 15^2 + 6^2
AB^2 = 225 + 36
AB^2 = 261
AB = sqrt(261)

Now, we can use the Law of Sines to find the value of PC:

a/sin(A) = b/sin(B) = c/sin(C)

AB/sin(C) = PB/sin(A)
sqrt(261)/sin(C) = 6/sin(120°)

To find sin(120°), we can use the fact that sin(120°) = sin(180° - 120°) = sin(60°). The value of sin(60°) is sqrt(3)/2.

sqrt(261)/sin(C) = 6/(sqrt(3)/2)
sqrt(261)/(sin(C)) = 12/(sqrt(3))

Cross-multiplying, we get:

(sqrt(261) * sqrt(3)) / sin(C) = 12

Now, we can solve for sin(C):

sin(C) = (sqrt(261) * sqrt(3))/12

Using a calculator, we get sin(C) ≈ 0.778

Now, we can find the value of PC using the Law of Sines:

PC/sin(120°) = AB/sin(C)

PC/(sqrt(3)/2) = sqrt(261)/0.778

Cross-multiplying and simplifying, we get:

PC ≈ (sqrt(3) * sqrt(261) * 2)/(0.778 * 2)
PC ≈ 12.003

Therefore, the value of PC is approximately 12.003.