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P R 18 m 8 m 25° Q

P R 18 m 8 m 25° Q

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

What is the area of triangle ΔPQR to the nearest tenth of a square meter? Drawing is not to scale.

A. 60.8m

B. 65.3m

C. 30.4m

D. 32.7m

### Understanding Angles and Lengths in Geometric Figures In this diagram, we examine a geometric figure composed of three points labeled P, Q, and R, forming a triangle. Here's a detailed description and analysis of this diagram: 1. **Triangle PQP:** - **Length PQ:** The side between points P and Q measures 18 meters. - **Length QR:** The side between points Q and R measures 8 meters. - **Angle PQR:** The angle at point Q between the lines PQ and QR is 25°. ### Detailed Breakdown: - **Vertices and Sides:** - **Point P:** A vertex of the triangle. - **Point Q:** A vertex of the triangle where the angle is known. - **Point R:** A vertex of the triangle. - **Measurements:** - **Side PQ:** 18 meters long. - **Side QR:** 8 meters long. - **Angle PQR:** An internal angle of 25°, located at vertex Q. When working with geometric figures like this, it’s essential to understand the relationships between side lengths and angles. This particular diagram can be useful for solving problems involving the Law of Sines or Cosines, which are fundamental concepts in trigonometry. ### Applications: - **Law of Sines:** A formula that relates the lengths of the sides of a triangle to the sines of its angles. - **Law of Cosines:** A formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. This specific geometric setup can be commonly encountered in various math problems and is critical for developing skills in geometry and trigonometry. ### Practice Problems: 1. **Problem 1:** Given the lengths of sides PQ and QR, and the angle PQR, can you determine the length of side PR using the Law of Cosines? 2. **Problem 2:** Using the given angle and sides, calculate the area of the triangle using trigonometric methods. Understanding these principles can significantly enhance your problem-solving capability in mathematics.
Transcribed Image Text:### Understanding Angles and Lengths in Geometric Figures In this diagram, we examine a geometric figure composed of three points labeled P, Q, and R, forming a triangle. Here's a detailed description and analysis of this diagram: 1. **Triangle PQP:** - **Length PQ:** The side between points P and Q measures 18 meters. - **Length QR:** The side between points Q and R measures 8 meters. - **Angle PQR:** The angle at point Q between the lines PQ and QR is 25°. ### Detailed Breakdown: - **Vertices and Sides:** - **Point P:** A vertex of the triangle. - **Point Q:** A vertex of the triangle where the angle is known. - **Point R:** A vertex of the triangle. - **Measurements:** - **Side PQ:** 18 meters long. - **Side QR:** 8 meters long. - **Angle PQR:** An internal angle of 25°, located at vertex Q. When working with geometric figures like this, it’s essential to understand the relationships between side lengths and angles. This particular diagram can be useful for solving problems involving the Law of Sines or Cosines, which are fundamental concepts in trigonometry. ### Applications: - **Law of Sines:** A formula that relates the lengths of the sides of a triangle to the sines of its angles. - **Law of Cosines:** A formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. This specific geometric setup can be commonly encountered in various math problems and is critical for developing skills in geometry and trigonometry. ### Practice Problems: 1. **Problem 1:** Given the lengths of sides PQ and QR, and the angle PQR, can you determine the length of side PR using the Law of Cosines? 2. **Problem 2:** Using the given angle and sides, calculate the area of the triangle using trigonometric methods. Understanding these principles can significantly enhance your problem-solving capability in mathematics.
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