Question

how many triangles exist with the given side lengths? 7 in, 7 in, 7 in a. no triangle exists with the given side lengths. b. more than one triangle exists with the given side lengths. c. exactly one unique triangle exists with the given side lengths.

Explanation:

Step1: Recall triangle - side rule

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For side lengths \(a = 7\), \(b = 7\), and \(c = 7\), we have \(a + b=7 + 7=14>7\), \(a + c=7 + 7 = 14>7\), and \(b + c=7+7 = 14>7\). So, a triangle can be formed.

Step2: Consider uniqueness

By the Side - Side - Side (SSS) congruence criterion, if the three side lengths of a triangle are fixed, then the triangle is unique. Since all three side lengths are \(7\) inches, exactly one unique equilateral triangle can be formed.

Answer:

C. Exactly one unique triangle exists with the given side lengths.