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a 60° h с 30° b If side c measures 30.5 units long, how long is side a?

a 60° h с 30° b If side c measures 30.5 units long, how long is side a?

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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### Geometry Problem: Solving for the Length of Side a #### Problem Statement: Consider the right triangle shown below: **Triangle Specifications:** - The triangle has one angle measuring 60° and another angle measuring 30°. - Side \(c\) is the hypotenuse. - Side \(a\) is opposite the 60° angle. - Side \(b\) is opposite the 30° angle. - Side \(h\) is an altitude drawn from the 60° angle to the hypotenuse. **Given Data:** - Length of side \(c\) (hypotenuse) = 30.5 units. **Question:** How long is side \(a\)? **Diagram Description:** - The right triangle is oriented such that: - The 60° angle is at the top-left vertex. - The 30° angle is at the bottom-right vertex. - The 90° angle is at the bottom-left vertex. - Side \(a\) is vertical, extending from the bottom-left vertex to the top-left vertex. - Side \(b\) is horizontal, extending from the bottom-left vertex to the bottom-right vertex. - Side \(c\), the hypotenuse, extends from the bottom-right vertex to the top-left vertex. - An altitude \(h\) is drawn from the 60° angle to the hypotenuse, forming two smaller right triangles within the original triangle, each having the right angle at the point where altitude \(h\) meets the hypotenuse. ### Solution Box: <input type="text" placeholder="Enter the length of side a here"> --- **Educational Note:** The provided information describes a 30-60-90 triangle, a special right triangle with inherent properties relating to the lengths of its sides. The ratios of the sides for such a triangle are fixed: the side opposite the 30° angle is half the hypotenuse, and the side opposite the 60° angle is \( \frac{\sqrt{3}}{2} \) times the hypotenuse. This information can be used to solve for side \(a\).
Transcribed Image Text:### Geometry Problem: Solving for the Length of Side a #### Problem Statement: Consider the right triangle shown below: **Triangle Specifications:** - The triangle has one angle measuring 60° and another angle measuring 30°. - Side \(c\) is the hypotenuse. - Side \(a\) is opposite the 60° angle. - Side \(b\) is opposite the 30° angle. - Side \(h\) is an altitude drawn from the 60° angle to the hypotenuse. **Given Data:** - Length of side \(c\) (hypotenuse) = 30.5 units. **Question:** How long is side \(a\)? **Diagram Description:** - The right triangle is oriented such that: - The 60° angle is at the top-left vertex. - The 30° angle is at the bottom-right vertex. - The 90° angle is at the bottom-left vertex. - Side \(a\) is vertical, extending from the bottom-left vertex to the top-left vertex. - Side \(b\) is horizontal, extending from the bottom-left vertex to the bottom-right vertex. - Side \(c\), the hypotenuse, extends from the bottom-right vertex to the top-left vertex. - An altitude \(h\) is drawn from the 60° angle to the hypotenuse, forming two smaller right triangles within the original triangle, each having the right angle at the point where altitude \(h\) meets the hypotenuse. ### Solution Box: <input type="text" placeholder="Enter the length of side a here"> --- **Educational Note:** The provided information describes a 30-60-90 triangle, a special right triangle with inherent properties relating to the lengths of its sides. The ratios of the sides for such a triangle are fixed: the side opposite the 30° angle is half the hypotenuse, and the side opposite the 60° angle is \( \frac{\sqrt{3}}{2} \) times the hypotenuse. This information can be used to solve for side \(a\).
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