Question

question 8 of 10
a triangle has two sides of lengths 4 and 15. what value could the length of the third side be? check all that apply.
a. 13
b. 15
c. 19
d. 12
e. 11
f. 4

Explanation:

Step1: Determine the range of the third side

Let the two sides of the triangle be \(a = 4\) and \(b=15\). 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 \(|a - b|\): \(|4-15|=11\). Then calculate \(a + b\): \(4 + 15=19\). So, \(11\lt c\lt19\).

Step2: Check each option

• Option A:

Since \(11\lt13\lt19\), \(13\) satisfies the triangle - inequality theorem.

• Option B:

Since \(11\lt15\lt19\), \(15\) satisfies the triangle - inequality theorem.

• Option C:

Since \(c = 19\) does not satisfy \(c\lt19\) (it should be \(c\lt19\) from \(a + b=19\)), \(19\) does not satisfy the triangle - inequality theorem.

• Option D:

Since \(11\lt12\lt19\), \(12\) satisfies the triangle - inequality theorem.

• Option E:

Since \(c = 11\) does not satisfy \(c\gt11\) (it should be \(c\gt11\) from \(|a - b| = 11\)), \(11\) does not satisfy the triangle - inequality theorem.

• Option F:

Since \(c = 4\) does not satisfy \(c\gt11\), \(4\) does not satisfy the triangle - inequality theorem.

Answer:

A. 13, B. 15, D. 12