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Let AABC be any right triangle (see figure below), with the measures of the sides as indicated and the altitude to the hypotenuse of length h. Observe the triangles formed by drawing this altitude, CD, which are separated and rearranged into comparable positions. B Z CL C D b C2 1. How are the three triangles related? How do you know? Original triangle b Subtriangle h #1 C1 Subhange C₂ h 2. Determine all the equivalent ratios between the side lengths for the original triangle and sub-triangle #1. 3. Determine all the equivalent ratios between the side lengths for the original triangle and sub-triangle #2. 4. Use some of the ratios you found in #2 and #3 to derive the Pythagorean Theorem, e.g., a² + b² = c².

Let AABC be any right triangle (see figure below), with the measures of the sides as indicated and the altitude to the hypotenuse of length h. Observe the triangles formed by drawing this altitude, CD, which are separated and rearranged into comparable positions. B Z CL C D b C2 1. How are the three triangles related? How do you know? Original triangle b Subtriangle h #1 C1 Subhange C₂ h 2. Determine all the equivalent ratios between the side lengths for the original triangle and sub-triangle #1. 3. Determine all the equivalent ratios between the side lengths for the original triangle and sub-triangle #2. 4. Use some of the ratios you found in #2 and #3 to derive the Pythagorean Theorem, e.g., a² + b² = c².

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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**The Pythagorean Theorem from Similar Right Triangles** Let ΔABC be any right triangle (see figure below), with the measures of the sides as indicated and the altitude to the hypotenuse of length \(h\). Observe the triangles formed by drawing this altitude, \(CD\), which are separated and rearranged into comparable positions. **Diagram Explanation:** - The left diagram shows ΔABC as a right triangle with: - Hypotenuse \(c\), - Right angle at \(C\), - Segment \(CD\) as the altitude from \(C\) to the hypotenuse \(AB\). - The hypotenuse is divided into segments \(c_1\) and \(c_2\) by point \(D\). - The middle and right sections rearrange the original triangle into three separate triangles: - **Original triangle**: The same as ΔABC, labeled with sides \(a\), \(b\), and \(c\). - **Subtriangle #1**: On the right, shows a smaller right triangle with sides \(a\), \(c_1\), and \(h\). - **Subtriangle #2**: Below Subtriangle #1, shows another smaller right triangle with sides \(b\), \(c_2\), and \(h\). **Questions:** 1. **How are the three triangles related? How do you know?** 2. **Determine all the equivalent ratios between the side lengths for the original triangle and sub-triangle #1.** 3. **Determine all the equivalent ratios between the side lengths for the original triangle and sub-triangle #2.** 4. **Use some of the ratios you found in #2 and #3 to derive the Pythagorean Theorem, e.g., \(a^2 + b^2 = c^2\).**
Transcribed Image Text:**The Pythagorean Theorem from Similar Right Triangles** Let ΔABC be any right triangle (see figure below), with the measures of the sides as indicated and the altitude to the hypotenuse of length \(h\). Observe the triangles formed by drawing this altitude, \(CD\), which are separated and rearranged into comparable positions. **Diagram Explanation:** - The left diagram shows ΔABC as a right triangle with: - Hypotenuse \(c\), - Right angle at \(C\), - Segment \(CD\) as the altitude from \(C\) to the hypotenuse \(AB\). - The hypotenuse is divided into segments \(c_1\) and \(c_2\) by point \(D\). - The middle and right sections rearrange the original triangle into three separate triangles: - **Original triangle**: The same as ΔABC, labeled with sides \(a\), \(b\), and \(c\). - **Subtriangle #1**: On the right, shows a smaller right triangle with sides \(a\), \(c_1\), and \(h\). - **Subtriangle #2**: Below Subtriangle #1, shows another smaller right triangle with sides \(b\), \(c_2\), and \(h\). **Questions:** 1. **How are the three triangles related? How do you know?** 2. **Determine all the equivalent ratios between the side lengths for the original triangle and sub-triangle #1.** 3. **Determine all the equivalent ratios between the side lengths for the original triangle and sub-triangle #2.** 4. **Use some of the ratios you found in #2 and #3 to derive the Pythagorean Theorem, e.g., \(a^2 + b^2 = c^2\).**
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