# Nathan cut an isosceles triangle from felt to make a spirit banner. Two sides of his banner had the following measures : 15 inches and 7 inches. Which could be the measure of the third side of Nathan’s banner ? A. 20 b. 15 c. 10 d. 7

Mar 22, 2018

B. $15$ inches

#### Explanation:

First off, let's list what we know about triangles:

1. In an isosceles triangle, two of the sides must be the same.
2. Any one side of a triangle cannot be longer than the sum of the other two sides (for example, if two sides of the triangle are $3$ and $5$, the third side length cannot be $12$, since there wouldn't be any possible way to connect those sides).

We know that two of the sides of the given triangle are $15$ and $7$. Since the triangle is isosceles, the other side must be either $15$ or $7$ (that way, there are two sides of the same length). Therefore, we can eliminate choice A $\left(20\right)$ and choice C $\left(10\right)$.

Now, let's look at our other choices. The three sides of the triangle must either be $15$, $15$, and $7$ or they must be $15$, $7$, and $7$.

If we assume that the triangle's sides are $15$, $7$, and $7$, then we have a problem: the longest side $\left(15\right)$ is bigger than the other two sides combined $\left(7 + 7 = 14\right)$. If you tried to construct an actual triangle with sides $15$, $7$ and $7$, you couldn't do it, because the longest side would push the other two sides so far apart that they couldn't touch each other!

This means that our only option is $15$, $15$, and $7$. If you check each side, you find out that none of the sides are larger than the sum of the other two sides! This means that this triangle CAN actually be constructed, and therefore it is the correct answer.

Therefore, the correct answer is B: the third side of Nathan's banner has a measure of $15$ inches.