Question

Given three segments of length x, 11-x, and x-4, respectively. Which of the following indicates the set of all numbers x such that the 3 segments could be the lengths of the sides of a triangle?

Answer 1

Answer choices : A) x>4 B) x<11 C) 0<x<7 D) 5<x<15 E) 5<x<7

Answer 2

Try each of them:

A) Using 4 as a starting point, lengths would be 4, 7, and 0. Not going to be possible (even if you use 4.1 or so).

B) Using 11 as a starting point, lengths would be 11, 0, and 7. Now as you go down less than 11, the first number (x) will get smaller and eventually go to zero and so will the last number (x-4) so this is not a good answer also.

C) Using 0 as our starting point, we get 0, 11, and -4. Even if we use say 1, it will be 1, 10 and -3 which is not possible since a triangle can't have a negative length, so it is wrong also.

D) Using 5 as our starting point, lengths would be 5, 6, and 1. We know that the two lengths of the smaller sides MUST add up to MORE than the length of the third side (the hypotenuse). With this, 5+1 equals six but is not greater than it, so we would say this is wrong too BUT it says only numbers greater than 5, and if we use 5.1 it would work so the 5 is okay. Let's check the other side of the constraint. If we use 15 (since it is the highest possible number) we get for the sides 15, -4 and 11. Even if we were to make that say 14 instead of 15 that second number would still be negative and so this answer must be wrong.

E) This is obviously the right answer because all others are wrong, but here is why. The greater than 5 already works, we just checked that in part d. Now, let's check the 7-the lengths would be 7, 4, and 3. The two smaller sides 3+4 equal 7 but are not greater than it. However as long as we stay below 7 (since it is 5<x<7) the sum of those two smaller sides will be greater than the third side. This is the right answer.

Hope this helps.
-Gabe

Answer 3

I was looking for an easier way to solve it but thanks it does help a lot :)

Answer 4

Well, basically, the first two answers are clearly wrong off the bat by simple thinking. The last two you have to look at a little closer. Thanks for the medal.