11. Problem number 11. A triangular pyramid has edges of integer length. Four of these lengths are as shown in the diagram. What is the sum of the lengths of the other two edges? First, let's take this diagram and label the other two edges. We'll call them A and B. The triangle inequality states that the sum of any two sides of a triangle are greater than the third, if we have a non-degenerate triangle. We could have the two sides of a triangle equal to the third, but that would just be a straight line, so we want the two sides of a triangle to be greater than the third side. So, let's apply that inequality into places, into triangles all over this diagram. For example, for triangle ABD, we have that 7-3 is less than A, because A plus 3 should be greater than 7, and A must be less than the sum of the other two sides, AB and AD, which is 7 plus 3. So, simplifying, 4 is less than A is less than 10. From triangle BCD, we have 4-2 is less than A, because A plus 2 must be greater than 4, and A is less than the sum of the other two sides, so A is less than 4 plus 2. Simplifying, A is between 2 and 6. So, if A is greater than 4, and A is less than 6, well, the only integer that satisfies that is A equals 5. From triangle ABC, we have that 7-2 is less than B is less than 7 plus 2. So, 5 is less than B is less than 9. But, that's not enough information, because at this point B could be equal to 6, 7, or 8. So, we also use triangle ACD. From triangle ACD, we have that B plus 3 must be greater than 4, so 4-3 is less than B, and B is less than the sum of the other two sides, which is 4 plus 3. So, 1 is less than B is less than 7, and B is equal to 6. That means that we have figured out the two sides, A equals 5 and B equals 6. Their sum gives us our answer of 11. So, the correct answer is choice C.

triangular pyramid

integer lengths

triangle inequality

edge lengths

solution