Question

angles and triangles - unit test a

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question 13

1. various pieces of string are cut at different lengths. stephanie selects three pieces that measure 4.5 cm, 6 cm, and 8.5 cm. determine what type of triangle stephanie will form.

○ a unique triangle
○ more than one triangle
○ no triangle
○ a right triangle

Explanation:

Step1: Check triangle inequality

For three side lengths \(a\), \(b\), \(c\) (where \(c\) is the longest side), the triangle inequality states \(a + b>c\). Here, \(a = 4.5\), \(b = 6\), \(c = 8.5\). Calculate \(4.5+6=10.5\), and \(10.5>8.5\), so a triangle can be formed.

Step2: Check for unique triangle

When three side lengths are given (SSS - Side - Side - Side), by the SSS congruence criterion, there is exactly one unique triangle that can be formed with those side lengths. Also, check if it's a right triangle: \(4.5^{2}+6^{2}=20.25 + 36=56.25\), and \(8.5^{2}=72.25\). Since \(56.25
eq72.25\), it's not a right triangle. And since SSS gives a unique triangle, it's a unique triangle.

Answer:

a unique triangle