Question

question 7 of 10
a triangle has two sides of lengths 7 and 12. what value could the length of the third side be? check all that apply.
a. 11
b. 5
c. 9
d. 7
e. 3
f. 17

Explanation:

Step1: Apply triangle inequality theorem

Let the two sides of the triangle be \(a = 7\) and \(b=12\). According to the triangle - inequality theorem, for a triangle with side lengths \(a\), \(b\), and \(c\), \(|a - b|\lt c\lt a + b\).
First, calculate \(|a - b|\) and \(a + b\).
\(|7-12|=| - 5| = 5\) and \(7 + 12=19\). So, \(5\lt c\lt19\).

Step2: Check each option

• Option A:

For \(c = 11\), since \(5\lt11\lt19\), \(11\) satisfies the inequality.

• Option B:

For \(c = 5\), since \(c = 5\) does not satisfy \(5\lt c\) (it should be \(5\lt c\), not \(5=c\)), \(5\) does not satisfy the inequality.

• Option C:

For \(c = 9\), since \(5\lt9\lt19\), \(9\) satisfies the inequality.

• Option D:

For \(c = 7\), since \(5\lt7\lt19\), \(7\) satisfies the inequality.

• Option E:

For \(c = 3\), since \(3\lt5\), \(3\) does not satisfy the inequality.

• Option F:

For \(c = 17\), since \(5\lt17\lt19\), \(17\) satisfies the inequality.

Answer:

A. 11, C. 9, D. 7, F. 17