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14) Find the area of the triangle below... A. 478.2 in² B. 298.9 in² 20 in 85° 30 in 24 in C. 239.1 in² 358.6 in²

14) Find the area of the triangle below... A. 478.2 in² B. 298.9 in² 20 in 85° 30 in 24 in C. 239.1 in² 358.6 in²

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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### Finding the Area of a Triangle Given the problem: **Find the area of the triangle shown below:** The triangle provided has the following measurements: - One angle: 85° - Two sides: 20 inches and 30 inches **Multiple Choice Answers:** - A. 478.2 in² - B. 298.9 in² - C. 239.1 in² - D. 358.6 in² (marked as incorrect with a red cross) **Diagram Explanation:** The triangle is non-right-angled, and the given measurements suggest using trigonometry for calculating the area. Specifically, the Area (A) can be calculated using the formula: \[ \text{Area} = \frac{1}{2} \times a \times b \times \sin(C) \] where \(a\) and \(b\) are the lengths of two sides, and \(C\) is the included angle between these sides. In this problem: - \( a = 20 \text{ in} \) - \( b = 30 \text{ in} \) - \( C = 85° \) Plugging these values into the formula: \[ \text{Area} = \frac{1}{2} \times 20 \times 30 \times \sin(85°) \] Using a calculator to find \(\sin(85°)\): \[ \sin(85°) \approx 0.9962 \] Thus: \[ \text{Area} = \frac{1}{2} \times 20 \times 30 \times 0.9962 \approx 298.86 \text{ in}² \] Since the choice B (298.9 in²) is closest to the computed area, it is the correct answer. ### Summary - The correct area of the triangle is approximately 298.9 in². - Confirmed as correct based on the closeness to theoretical calculation.
Transcribed Image Text:### Finding the Area of a Triangle Given the problem: **Find the area of the triangle shown below:** The triangle provided has the following measurements: - One angle: 85° - Two sides: 20 inches and 30 inches **Multiple Choice Answers:** - A. 478.2 in² - B. 298.9 in² - C. 239.1 in² - D. 358.6 in² (marked as incorrect with a red cross) **Diagram Explanation:** The triangle is non-right-angled, and the given measurements suggest using trigonometry for calculating the area. Specifically, the Area (A) can be calculated using the formula: \[ \text{Area} = \frac{1}{2} \times a \times b \times \sin(C) \] where \(a\) and \(b\) are the lengths of two sides, and \(C\) is the included angle between these sides. In this problem: - \( a = 20 \text{ in} \) - \( b = 30 \text{ in} \) - \( C = 85° \) Plugging these values into the formula: \[ \text{Area} = \frac{1}{2} \times 20 \times 30 \times \sin(85°) \] Using a calculator to find \(\sin(85°)\): \[ \sin(85°) \approx 0.9962 \] Thus: \[ \text{Area} = \frac{1}{2} \times 20 \times 30 \times 0.9962 \approx 298.86 \text{ in}² \] Since the choice B (298.9 in²) is closest to the computed area, it is the correct answer. ### Summary - The correct area of the triangle is approximately 298.9 in². - Confirmed as correct based on the closeness to theoretical calculation.
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