Question

question 10 of 10
a triangle has two sides of lengths 8 and 10. what value could the length of
the third side be? check all that apply.

a. 20
b. 18
c. 8
d. 7
e. 4
f. 10

Explanation:

Step1: Recall Triangle Inequality Theorem

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side, and the difference of the lengths of any two sides must be less than the length of the remaining side. For a triangle with sides \(a\), \(b\), and \(c\), we have \(|a - b| < c < a + b\).

Step2: Calculate the range for the third side

Let \(a = 8\) and \(b = 10\). First, find the difference: \(|10 - 8| = 2\). Then, find the sum: \(10 + 8 = 18\). So the length of the third side \(c\) must satisfy \(2 < c < 18\).

Step3: Check each option

• Option A: \(20\) is not less than \(18\), so it does not satisfy the inequality.
• Option B: \(18\) is not less than \(18\) (it's equal), so it does not satisfy the inequality.
• Option C: \(8\) is between \(2\) and \(18\) (\(2 < 8 < 18\)), so it satisfies the inequality.
• Option D: \(7\) is between \(2\) and \(18\) (\(2 < 7 < 18\)), so it satisfies the inequality.
• Option E: \(4\) is between \(2\) and \(18\) (\(2 < 4 < 18\)), so it satisfies the inequality.
• Option F: \(10\) is between \(2\) and \(18\) (\(2 < 10 < 18\)), so it satisfies the inequality.

Answer:

C. 8, D. 7, E. 4, F. 10