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The triangle with vertices R(- 2,3), S(2, – 1), and T(4,5) is shown to the right. The lengths of the three sides are computed below. RS = (-2-2)² + (3-(- 1))² = /16+16 = /32 RT = (-2- 4)² + (3 – 5)² V(2- 4² + (-1- 5)² = V36+ 4 = V40 /4+ 36 = V40 ST = %3D Is the triangle equilateral?

The triangle with vertices R(- 2,3), S(2, – 1), and T(4,5) is shown to the right. The lengths of the three sides are computed below. RS = (-2-2)² + (3-(- 1))² = /16+16 = /32 RT = (-2- 4)² + (3 – 5)² V(2- 4² + (-1- 5)² = V36+ 4 = V40 /4+ 36 = V40 ST = %3D Is the triangle equilateral?

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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**Question:** Determine whether the triangle is equilateral based on the given information: **Options:** - **A.** Because \(\sqrt{40} \neq \sqrt{32}\), not all sides are equal. Therefore, the triangle is not equilateral. - **B.** Because \(\sqrt{40} \neq \sqrt{32}\), it follows that no two angles are equal. Therefore, the triangle is not equilateral. - **C.** Because there are two sides of length \(\sqrt{40}\), two sides are equal. Therefore, the triangle is equilateral. - **D.** Because there are two sides of length \(\sqrt{40}\), it follows that two angles are equal. Therefore, the triangle is equilateral. **Analysis:** In this problem, you are tasked with assessing the conditions under which a triangle can be considered equilateral. An equilateral triangle requires all three sides to be of equal length. Examine the given options and the mathematical operations involved, such as evaluating the square roots and comparisons between different side lengths. Based on the provided options, it is necessary to determine if any of them logically conclude that the triangle can or cannot be equilateral.
Transcribed Image Text:**Question:** Determine whether the triangle is equilateral based on the given information: **Options:** - **A.** Because \(\sqrt{40} \neq \sqrt{32}\), not all sides are equal. Therefore, the triangle is not equilateral. - **B.** Because \(\sqrt{40} \neq \sqrt{32}\), it follows that no two angles are equal. Therefore, the triangle is not equilateral. - **C.** Because there are two sides of length \(\sqrt{40}\), two sides are equal. Therefore, the triangle is equilateral. - **D.** Because there are two sides of length \(\sqrt{40}\), it follows that two angles are equal. Therefore, the triangle is equilateral. **Analysis:** In this problem, you are tasked with assessing the conditions under which a triangle can be considered equilateral. An equilateral triangle requires all three sides to be of equal length. Examine the given options and the mathematical operations involved, such as evaluating the square roots and comparisons between different side lengths. Based on the provided options, it is necessary to determine if any of them logically conclude that the triangle can or cannot be equilateral.
**Triangle Analysis and Side Length Calculation** The triangle with vertices R(-2, 3), S(2, -1), and T(4, 5) is shown in the accompanying graph. The lengths of the three sides are computed below using the distance formula. **Distance Calculations:** - **RS:** \[ RS = \sqrt{(-2 - 2)^2 + (3 - (-1))^2} = \sqrt{(-4)^2 + (4)^2} = \sqrt{16 + 16} = \sqrt{32} \] - **RT:** \[ RT = \sqrt{(-2 - 4)^2 + (3 - 5)^2} = \sqrt{(-6)^2 + (-2)^2} = \sqrt{36 + 4} = \sqrt{40} \] - **ST:** \[ ST = \sqrt{(2 - 4)^2 + (-1 - 5)^2} = \sqrt{(-2)^2 + (-6)^2} = \sqrt{4 + 36} = \sqrt{40} \] **Analysis:** The calculations confirm that the triangle is not equilateral since the side lengths are not all equal (\(\sqrt{32}, \sqrt{40}, \sqrt{40}\)). **Graph Description:** The graph shows a coordinate plane with a triangle plotted. The vertices are marked at R(-2, 3), S(2, -1), and T(4, 5). The axes are labeled \(x\) and \(y\), with gridlines for visual reference. The triangle is highlighted in blue to emphasize its shape and orientation.
Transcribed Image Text:**Triangle Analysis and Side Length Calculation** The triangle with vertices R(-2, 3), S(2, -1), and T(4, 5) is shown in the accompanying graph. The lengths of the three sides are computed below using the distance formula. **Distance Calculations:** - **RS:** \[ RS = \sqrt{(-2 - 2)^2 + (3 - (-1))^2} = \sqrt{(-4)^2 + (4)^2} = \sqrt{16 + 16} = \sqrt{32} \] - **RT:** \[ RT = \sqrt{(-2 - 4)^2 + (3 - 5)^2} = \sqrt{(-6)^2 + (-2)^2} = \sqrt{36 + 4} = \sqrt{40} \] - **ST:** \[ ST = \sqrt{(2 - 4)^2 + (-1 - 5)^2} = \sqrt{(-2)^2 + (-6)^2} = \sqrt{4 + 36} = \sqrt{40} \] **Analysis:** The calculations confirm that the triangle is not equilateral since the side lengths are not all equal (\(\sqrt{32}, \sqrt{40}, \sqrt{40}\)). **Graph Description:** The graph shows a coordinate plane with a triangle plotted. The vertices are marked at R(-2, 3), S(2, -1), and T(4, 5). The axes are labeled \(x\) and \(y\), with gridlines for visual reference. The triangle is highlighted in blue to emphasize its shape and orientation.
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