Two sides of a triangle have lengths 10 and 15. what must be true about the length of the third side?

A widely-known theorem can be used for this, but only if it is a right triangle (let's assume it is).

Pythagorean Theorem, or a^2+b^2=c^2 (the ^ means it is brought up to that power, so it would be a(squared) + b(squared) = c(squared))

Let's plug in our numbers into this equation to see what we get,

(Assuming that 10 and 15 are the legs of the triangle.)
10^2 + 15^2 = c^2 (c is your hypotenuse, or your longest side of the triangle.)

10^2 = 100  &  15^2 =  225
100 + 225 = c^2
325 = c^2
(square root)325 = c
18.03 ≈ c

So, your final equation should look like this:

10^2 + 15^2 = 18.03^2
Which it does! 100 + 225 = 325

What must be true about the length of the third side?
The third side must equal 18.03, only if it is a right triangle.