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Triangle KJL is inscribed in semicircle O shown below such that JK = 6 and JL =11. Which of the following is closest to the radius of the semicircle? (1) 5.68 (2) 6.26 (3) 10.61 (4) 12.53 K

Triangle KJL is inscribed in semicircle O shown below such that JK = 6 and JL =11. Which of the following is closest to the radius of the semicircle? (1) 5.68 (2) 6.26 (3) 10.61 (4) 12.53 K

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 Explanation **Problem Statement:** Triangle \( KJL \) is inscribed in a semicircle \( O \) shown below such that \( JK = 6 \) and \( JL = 11 \). Which of the following is closest to the radius of the semicircle? 1. 5.68 2. 6.26 3. 10.61 4. 12.53 **Diagram Description:** The provided diagram illustrates a semicircle with center \( O \). A triangle \( KJL \) is inscribed within this semicircle. Here are the notable points and lengths: - Point \( J \) is on the circumference of the semicircle. - Points \( K \) and \( L \) are endpoints of the diameter of the semicircle, \( KL \). - Segment \( JK \) is given as 6 units. - Segment \( JL \) is given as 11 units. - Line segment \( JO \) is also a radius of the semicircle. **Step-by-Step Solution:** Using the properties of the inscribed triangle in a semicircle, the hypotenuse of the right triangle \( KJL \) is the diameter, and the triangle \( KJL \) is right-angled at \( J \). 1. By the Pythagorean theorem, we know: \[ KL = \sqrt{JK^2 + JL^2} = \sqrt{6^2 + 11^2} = \sqrt{36 + 121} = \sqrt{157} \] 2. The diameter of the semicircle is \( \sqrt{157} \). 3. The radius is half the diameter: \[ \text{Radius} = \frac{\sqrt{157}}{2} \approx \frac{12.53}{2} = 6.26 \] Therefore, the closest to the radius of the semicircle is: \[ (2) 6.26 \]
Transcribed Image Text:### Geometry Problem Explanation **Problem Statement:** Triangle \( KJL \) is inscribed in a semicircle \( O \) shown below such that \( JK = 6 \) and \( JL = 11 \). Which of the following is closest to the radius of the semicircle? 1. 5.68 2. 6.26 3. 10.61 4. 12.53 **Diagram Description:** The provided diagram illustrates a semicircle with center \( O \). A triangle \( KJL \) is inscribed within this semicircle. Here are the notable points and lengths: - Point \( J \) is on the circumference of the semicircle. - Points \( K \) and \( L \) are endpoints of the diameter of the semicircle, \( KL \). - Segment \( JK \) is given as 6 units. - Segment \( JL \) is given as 11 units. - Line segment \( JO \) is also a radius of the semicircle. **Step-by-Step Solution:** Using the properties of the inscribed triangle in a semicircle, the hypotenuse of the right triangle \( KJL \) is the diameter, and the triangle \( KJL \) is right-angled at \( J \). 1. By the Pythagorean theorem, we know: \[ KL = \sqrt{JK^2 + JL^2} = \sqrt{6^2 + 11^2} = \sqrt{36 + 121} = \sqrt{157} \] 2. The diameter of the semicircle is \( \sqrt{157} \). 3. The radius is half the diameter: \[ \text{Radius} = \frac{\sqrt{157}}{2} \approx \frac{12.53}{2} = 6.26 \] Therefore, the closest to the radius of the semicircle is: \[ (2) 6.26 \]
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