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Which sets of measurements could be the side lengths of a triangle? Select each correct answer. A \( 4 \mathrm{~cm}, 8 \mathrm{~cm}, 13 \mathrm{~cm} \) B \( 3 \mathrm{~cm}, 3 \mathrm{~cm}, 3 \mathrm{~cm} \) C \( 6 \mathrm{~cm}, 7 \mathrm{~cm}, 8 \mathrm{~cm} \) D a \( 7 \mathrm{~cm}, 7 \mathrm{~cm}, 10 \mathrm{~cm} \) E \( 5 \mathrm{~cm}, 9 \mathrm{~cm}, 14 \mathrm{~cm} \)

Ask by Sanders Todd. in the United States
Jan 24,2025

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Answer

The correct answers are B, C, and D.

Solution

To determine which sets of measurements can be the side lengths of a triangle, we use the **Triangle Inequality Theorem**. This theorem states that for any triangle, the sum of the lengths of any two sides must be **greater** than the length of the remaining side. Let's evaluate each option: - **A. \(4 \text{ cm}, 8 \text{ cm}, 13 \text{ cm}\)** - \(4 + 8 = 12 \), and \(12\) is **not greater** than \(13\). - **Not a valid triangle.** - **B. \(3 \text{ cm}, 3 \text{ cm}, 3 \text{ cm}\)** - \(3 + 3 = 6 > 3\) - All combinations satisfy the inequality. - **Valid triangle.** - **C. \(6 \text{ cm}, 7 \text{ cm}, 8 \text{ cm}\)** - \(6 + 7 = 13 > 8\) - \(6 + 8 = 14 > 7\) - \(7 + 8 = 15 > 6\) - **Valid triangle.** - **D. \(7 \text{ cm}, 7 \text{ cm}, 10 \text{ cm}\)** - \(7 + 7 = 14 > 10\) - \(7 + 10 = 17 > 7\) - \(7 + 10 = 17 > 7\) - **Valid triangle.** - **E. \(5 \text{ cm}, 9 \text{ cm}, 14 \text{ cm}\)** - \(5 + 9 = 14\), and \(14\) is **not greater** than \(14\). - **Not a valid triangle.** **Therefore, the sets of measurements that could be the side lengths of a triangle are:** **B, C, and D**

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To determine whether sets of measurements can form a triangle, you can use the Triangle Inequality Theorem. This theorem states that the sum of any two side lengths must be greater than the third side length. So, let's break down the options and see which sets fit the bill! B \(3 \text{ cm}, 3 \text{ cm}, 3 \text{ cm}\): All sides are equal, so they certainly form an equilateral triangle! C \(6 \text{ cm}, 7 \text{ cm}, 8 \text{ cm}\): The sum of every pair of sides is greater than the third side (6+7 > 8, 6+8 > 7, 7+8 > 6), making this a valid triangle as well! D \(7 \text{ cm}, 7 \text{ cm}, 10 \text{ cm}\): This forms an isosceles triangle, as the two sides are equal and the Triangle Inequality Theorem holds. Now, A \(4 \text{ cm}, 8 \text{ cm}, 13 \text{ cm}\) and E \(5 \text{ cm}, 9 \text{ cm}, 14 \text{ cm}\) do not meet the requirement since \(4 + 8 = 12\) (less than 13) and \(5 + 9 = 14\) (not strictly greater). So, A and E fall flat in their quest to be triangles! In summary, the sets that can make triangles are B, C, and D—everyone else can hit the road!

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