Question 1090989: Hi, so I'm working on some geometry problems and this question has me really stuck, if you could help me, it would be great.
Problem:
The sides of an isosceles triangle are 17, 17, and 16. From any point, P, on the base, draw a line perpendicular to the base that intersects one of the equal sides of the triangle at R. Label the point Q where the line intersects an extension of the third side of the triangle. Find the sum of PQ+PR.
Thanks in advance.
Answer by greenestamps(13203)   (Show Source):
You can put this solution on YOUR website!
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.
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.
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.
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.
And, because of the way the problem is stated, that has to be the answer, regardless of what point we choose for P.
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.
Now for a formal derivation of that answer....
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.
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.
But then the length of PR is 15-x, and the length of PQ is 15+x; so the sum PQ+PR is 30.
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