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). 10 a = b = 6 8 b a

). 10 a = b = 6 8 b 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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This image features a mathematical problem involving right triangles. It depicts a right triangle alongside an inscribed right triangle. Detailed elements of the image and accompanying problem are as follows: ### Diagram Description: - There is a larger right-angle triangle with one side labeled as 10. - An inscribed right-angle triangle shares vertices with the larger triangle. The hypotenuse of this inscribed triangle is labeled as 8. - One of the legs of the smaller triangle is labeled as 6. - The shared sides opposite the right angles in both triangles are labeled 'a' and 'b'. ### Problem Statement: Below the diagram, there are two expressions requiring solutions: - \( a = \) - \( b = \) Each expression is followed by a dotted rectangular box where the solver is expected to fill in the values of \( a \) and \( b \). ### Educational Explanation: This problem can be solved using the Pythagorean theorem. In a right-angled triangle, the theorem states that the square of the hypotenuse (the triangle's longest side) is equal to the sum of the squares of the other two sides. #### Steps to Solve: 1. **Identify right-angled triangles:** - Larger triangle: Hypotenuse = 10. - Smaller inscribed triangle: Hypotenuse = 8, one leg = 6. 2. **Use the Pythagorean theorem to find the unknown side 'a':** \[ a^2 = 8^2 - 6^2 \] \[ a^2 = 64 - 36 \] \[ a^2 = 28 \] \[ a = \sqrt{28} \] \[ a = 2\sqrt{7} \] 3. **Use the Pythagorean theorem to find the unknown side 'b' of the larger triangle utilizing one leg as \( a = 2\sqrt{7} \) and hypotenuse 10:** \[ b^2 = 10^2 - (2\sqrt{7})^2 \] \[ b^2 = 100 - 28 \] \[ b^2 = 72 \] \[ b = \sqrt{72} \] \[ b
Transcribed Image Text:This image features a mathematical problem involving right triangles. It depicts a right triangle alongside an inscribed right triangle. Detailed elements of the image and accompanying problem are as follows: ### Diagram Description: - There is a larger right-angle triangle with one side labeled as 10. - An inscribed right-angle triangle shares vertices with the larger triangle. The hypotenuse of this inscribed triangle is labeled as 8. - One of the legs of the smaller triangle is labeled as 6. - The shared sides opposite the right angles in both triangles are labeled 'a' and 'b'. ### Problem Statement: Below the diagram, there are two expressions requiring solutions: - \( a = \) - \( b = \) Each expression is followed by a dotted rectangular box where the solver is expected to fill in the values of \( a \) and \( b \). ### Educational Explanation: This problem can be solved using the Pythagorean theorem. In a right-angled triangle, the theorem states that the square of the hypotenuse (the triangle's longest side) is equal to the sum of the squares of the other two sides. #### Steps to Solve: 1. **Identify right-angled triangles:** - Larger triangle: Hypotenuse = 10. - Smaller inscribed triangle: Hypotenuse = 8, one leg = 6. 2. **Use the Pythagorean theorem to find the unknown side 'a':** \[ a^2 = 8^2 - 6^2 \] \[ a^2 = 64 - 36 \] \[ a^2 = 28 \] \[ a = \sqrt{28} \] \[ a = 2\sqrt{7} \] 3. **Use the Pythagorean theorem to find the unknown side 'b' of the larger triangle utilizing one leg as \( a = 2\sqrt{7} \) and hypotenuse 10:** \[ b^2 = 10^2 - (2\sqrt{7})^2 \] \[ b^2 = 100 - 28 \] \[ b^2 = 72 \] \[ b = \sqrt{72} \] \[ b
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