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Given AADC and AAEB, What is AE? 72 A B 55 88- - AE %3D

Given AADC and AAEB, What is AE? 72 A B 55 88- - AE %3D

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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### Educational Website: Geometry Problem Explanation #### Problem Statement Given triangles \( \triangle ADC \) and \( \triangle AEB \), find the length of segment \( AE \). #### Diagram Description The provided diagram shows two right triangles, \( \triangle ADC \) and \( \triangle AEB \), arranged such that: - \( \triangle ADC \) is a larger right triangle with \( \angle D \) as the right angle. - \( \triangle AEB \) is a smaller right triangle within \( \triangle ADC \) with \( \angle B \) as the right angle. In the diagram: - Point \( A \) is the common vertex of both triangles. - Line segment \( AE \) is shared between the two triangles. Dimensions provided: - \( DC = 72 \) units - \( BC = 55 \) units - \( AB = 88 \) units ### Task Using the information given in the problem and the diagram, calculate the length of \( AE \). #### Solution Outline To find \( AE \), follow these steps: 1. **Identify the Relationships in the Triangles:** - Note that \( \triangle AEB \) is similar to \( \triangle ADC \) because they are both right triangles sharing an acute angle at \( A \). 2. **Use the Pythagorean Theorem:** - Apply the Pythagorean theorem to both triangles to find the missing side lengths. 3. **Solve for \( AE \):** - Utilize the properties of similar triangles to compare side lengths and solve for \( AE \). Let's walk through these steps in detail to find the length \( AE \).
Transcribed Image Text:### Educational Website: Geometry Problem Explanation #### Problem Statement Given triangles \( \triangle ADC \) and \( \triangle AEB \), find the length of segment \( AE \). #### Diagram Description The provided diagram shows two right triangles, \( \triangle ADC \) and \( \triangle AEB \), arranged such that: - \( \triangle ADC \) is a larger right triangle with \( \angle D \) as the right angle. - \( \triangle AEB \) is a smaller right triangle within \( \triangle ADC \) with \( \angle B \) as the right angle. In the diagram: - Point \( A \) is the common vertex of both triangles. - Line segment \( AE \) is shared between the two triangles. Dimensions provided: - \( DC = 72 \) units - \( BC = 55 \) units - \( AB = 88 \) units ### Task Using the information given in the problem and the diagram, calculate the length of \( AE \). #### Solution Outline To find \( AE \), follow these steps: 1. **Identify the Relationships in the Triangles:** - Note that \( \triangle AEB \) is similar to \( \triangle ADC \) because they are both right triangles sharing an acute angle at \( A \). 2. **Use the Pythagorean Theorem:** - Apply the Pythagorean theorem to both triangles to find the missing side lengths. 3. **Solve for \( AE \):** - Utilize the properties of similar triangles to compare side lengths and solve for \( AE \). Let's walk through these steps in detail to find the length \( AE \).
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