Sample Question #199 (mathematics – geometry)

[For some reason, triangle questions are popular at quant interviews, especially at hedge funds]

Can you prove the following facts about a triangle?

- The total length of any two sides of a triangle is larger than the third side
- The difference in length between any two sides is smaller than the third side
- The three angles must add up to 180 degrees
- In a right triangle,
*a*^{2}+*b*^{2}=*c*^{2}, where *a *and *b *are the two right sides.

(Comment: I believe the purpose of questions like this is to see if you can think, and recall stuff you learned a long, long time ago, on your foot)

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ANSWER

1. It’s probably sufficient to provide a visual proof.

2. Follows immediately from #1.

3. Many ways; one way is to say a triangle is half of a parallelogram.

4. This one is hard…

4. You can prove this one with a visual.

Draw a square and pick a point at the same spot on each line. Call the smaller section a and call the larger section b. Now, if you were to connect the dots, you would be left with 5 shapes inside of the square. 4 trianges with sides a and b and a square in the middle. We will label each side of the square, c. Now, let see what we have. The area of the total square is a^2 + 2ab + b^2, because it’s simply (a + b)^2. The inside is 4 * (.5ab) = 2ab, from the four triangles, and c^2 from the square. Let’s set up our equation: a^2 + 2ab + b^2 = 2ab + c^2. The proof follows.