Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN: 9781337614085
Author: Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher: Cengage,
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Concept explainers

Ratios
A ratio is a comparison between two numbers of the same kind. It represents how many times one number contains another. It also represents how small or large one number is compared to the other.
Trigonometric Ratios
Trigonometric ratios give values of trigonometric functions. It always deals with triangles that have one angle measuring 90 degrees. These triangles are right-angled. We take the ratio of sides of these triangles.
Question
**Geometry Problem: 45°-45°-90° Triangle** **Problem Statement:** The hypotenuse of a 45°-45°-90° triangle measures \(22\sqrt{2}\) units. **Diagram Explanation:** There is a right triangle displayed with two angles marked as 45° each, making it a 45°-45°-90° triangle. The hypotenuse of the triangle is labeled as \(22\sqrt{2}\) units. The legs of the triangle are marked with red lines indicating equal length, characteristic of a 45°-45°-90° triangle. **Question:** What is the length of one leg of the triangle? **Answer Options:** - \(11\) units - \(11\sqrt{2}\) units - \(22\) units - \(22\sqrt{2}\) units ### Detailed Solution: In a 45°-45°-90° triangle, the sides have a unique ratio: the legs are equal in length, and the hypotenuse is \(\sqrt{2}\) times the length of each leg. If the hypotenuse \(c\) is given as \(22\sqrt{2}\) units, we can find the length of each leg \(a\) using the following relationship: \[ c = a\sqrt{2} \] \[ 22\sqrt{2} = a\sqrt{2} \] By dividing both sides by \(\sqrt{2}\), we find: \[ a = \frac{22\sqrt{2}}{\sqrt{2}} = 22 \] Thus, each leg of the triangle is \(22\) units long. **Correct Answer:** - \(22\) units
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Transcribed Image Text:**Geometry Problem: 45°-45°-90° Triangle** **Problem Statement:** The hypotenuse of a 45°-45°-90° triangle measures \(22\sqrt{2}\) units. **Diagram Explanation:** There is a right triangle displayed with two angles marked as 45° each, making it a 45°-45°-90° triangle. The hypotenuse of the triangle is labeled as \(22\sqrt{2}\) units. The legs of the triangle are marked with red lines indicating equal length, characteristic of a 45°-45°-90° triangle. **Question:** What is the length of one leg of the triangle? **Answer Options:** - \(11\) units - \(11\sqrt{2}\) units - \(22\) units - \(22\sqrt{2}\) units ### Detailed Solution: In a 45°-45°-90° triangle, the sides have a unique ratio: the legs are equal in length, and the hypotenuse is \(\sqrt{2}\) times the length of each leg. If the hypotenuse \(c\) is given as \(22\sqrt{2}\) units, we can find the length of each leg \(a\) using the following relationship: \[ c = a\sqrt{2} \] \[ 22\sqrt{2} = a\sqrt{2} \] By dividing both sides by \(\sqrt{2}\), we find: \[ a = \frac{22\sqrt{2}}{\sqrt{2}} = 22 \] Thus, each leg of the triangle is \(22\) units long. **Correct Answer:** - \(22\) units
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