Question

A triangle has one side length of 12 inches and another of 8 inches. Describe all possible lengths of the third side. Show and explain your reasoning.

Answer 1

The third side must be smaller than 20 inches and greater than 4 inches

`4<c<20`

Explanation:

Let a, b and c be the lengths of triangle's sides. Then

`a+b>c\\ \\a+c>b\\ \\b+c>a`

Use this rule in your case. So, if a=12 and b=8, then

`12+8>c\\ \\12+c>8\\ \\8+c>12`

Hence, you get

`c<20\\ \\c>-4\\ \\c>4`

From these inequalities, you can state that

`4<c<20`

So, c must be smaller than 20 inches and greater than 4 inches.

Answer 2

The possible lengths are all the real numbers greater than 4 inches and less than 20 inches

Explanation:

we know that

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Let

c----> the length of the third side of triangle

Applying the triangle inequality theorem

1) 12+8 > c

20 > c

Rewrite

c < 20 in

2) 8+c > 12

c > 12-8

c < 4 in

therefore

4 in < c < 20 in

The possible lengths are all the real numbers greater than 4 inches and less than 20 inches