Question 1090989
<br>We can use the symmetry of the isosceles triangle and the Pythagorean Theorem to find that the altitude of the triangle is 15.  We will need that number later on.<br>
I will give a formal derivation of the answer later.  But before I do that, let me point out a very powerful problem solving concept that can make this problem easy.<br>
The statement of the problem says you can choose ANY point P on the base of the triangle.  That means you will get the same answer regardless of which point you choose.  So choose a point that makes the problem easy to solve.<br>
For this problem, if we choose P to be the foot of the altitude of the isosceles triangle, then we get a "limiting case", in which PQ and PR are both the altitude of the triangle.  Since we know the altitude is 15, the sum of PQ and PR is 30.<br>
And, because of the way the problem is stated, that has to be the answer, regardless of what point we choose for P.<br>
So remember that concept when you are working a problem where the statement of the problem allows you to choose a figure that makes solving the problem easy.<br>
Now for a formal derivation of that answer....<br>
Let T be the vertex of the isosceles triangle.  Draw segment ST perpendicular to PQ (parallel to the base of the triangle), where S is on PQ.<br>
Triangle RTQ has angles Q and R congruent and so is isosceles.  With ST perpendicular to PQ, that makes triangles QTS and RTS congruent.  That means segments RS and QS are the same length; call that length x.<br>
But then the length of PR is 15-x, and the length of PQ is 15+x; so the sum PQ+PR is 30.