Question

analyzing triangle side lengths
gemma wants to draw a triangle with side lengths of 4 inches, 12 inches, and 17 inches. which statement is true?
this triangle exists because the sum of any two side lengths is greater than the length of the third side.
this triangle exists because the sum of 4 and 12 is less than 17.
this triangle does not exist because the sum of any two side lengths is greater than the length of the third side.
this triangle does not exist because the sum of 4 and 12 is less than 17.

Explanation:

Step1: Recall Triangle Inequality Theorem

The triangle inequality theorem states that for a triangle with side lengths \(a\), \(b\), and \(c\), the sum of any two sides must be greater than the third side, i.e., \(a + b>c\), \(a + c>b\), and \(b + c>a\).

Step2: Check the given side lengths

Given side lengths are \(4\) inches, \(12\) inches, and \(17\) inches. Let's check the sum of \(4\) and \(12\) first: \(4+12 = 16\). Now, compare this sum with the third side \(17\). We see that \(16<17\), which violates the triangle inequality theorem. So, the triangle does not exist because the sum of \(4\) and \(12\) is less than \(17\).

Answer:

This triangle does not exist because the sum of 4 and 12 is less than 17.