Question

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 ?

Answer 1

The third side must be 15 inches.

Since the triangle is isosceles, two sides are the same length.  This gives us the choices of 7 inches or 15 inches.

The triangle inequality theorem tells us that the sum of the smaller two sides must be greater than the length of the third side.  If we have 7, 7 and 15, this does not work, because 7+7 is not greater than 15.  Thus the missing side must be 15 inches.

Answer 2

we know that

Since the triangle is isosceles, two sides have the same length.  This gives us the possibility to choose between $7$ or $15$ inches.

The Triangle Inequality Theorem states that the sum of any $2$ sides of a triangle must be greater than the measure of the third side

Applying the Triangle Inequality Theorem, three conditions must be satisfied

Let

a,b,c ------> the length sides of the triangle

$a+b > c$

$a+c > b$

$b+c > a$

Let's analyze the two possibilities

Possibility N 1

$a=7\ in\\b=7\ in\\c=15\ in$

$7+7 > 15$ -------> is not ok

$7+15 > 7$ -----> is ok

$7+15 > 7$ -----> is ok

so

the third side cannot be $7$ inches

Possibility N 2

$a=7\ in\\b=15\ in\\c=15\ in$

$7+15 > 15$ -------> is ok

$7+15 > 15$ -----> is ok

$15+15 > 7$ -----> is ok

so

the third side is $15$ inches

therefore

the answer is

the third side is $15$ inches