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В 90° 12 cm А

В 90° 12 cm А

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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### Circle Geometry - Right Triangle in a Circle #### Diagram Explanation The diagram illustrates a circle with a right triangle inscribed within it. The vertices of the right triangle are labeled as follows: - Point \(A\): Located on the circumference of the circle. - Point \(B\): Located on the circumference of the circle. - Point \(C\): The vertex of the right angle, also located on the circumference of the circle. A right triangle is formed with the right angle at vertex \(C\), positioned at the top of the circle. #### Details - The circle is outlined with a green circumference. - The hypotenuse \( \overline{AB} \) lies along the diameter of the circle. - The length of the base \( \overline{AC} \) is given as 12 cm. - Line segments \( \overline{AC} \) and \( \overline{BC} \) form a right angle (90°) at \(C\). - A shaded purple region is created by the area of the circle outside the triangle and underneath the hypotenuse. ### Key Concepts 1. **Right Triangle Properties**: - In a right triangle inscribed within a circle, the hypotenuse is always the diameter of the circle. This is a consequence of the inscribed angle theorem, which states that an angle subtended by a diameter is a right angle. 2. **Circle Geometry**: - The relationship between the right triangle and the circle can be used to solve various geometric problems, such as finding the circle's radius, the length of the other sides of the triangle, and areas related to both the triangle and the circle. #### Example Problem: Given \( \overline{AC} = 12 \, \text{cm} \), and knowing \(\angle ACB = 90°\), find the radius of the circle. **Solution**: 1. Identify that \(\overline{AB}\) is the diameter of the circle. 2. Use the Pythagorean theorem to find \(\overline{BC}\) if needed. For right triangles inscribed in a circle, the hypotenuse \( \overline{AB} \) can be directly used to determine the diameter. This diagram is a useful educational tool for visualizing the relationship between right triangles and circles, illustrating key geometric principles, and solving related mathematical problems.
Transcribed Image Text:### Circle Geometry - Right Triangle in a Circle #### Diagram Explanation The diagram illustrates a circle with a right triangle inscribed within it. The vertices of the right triangle are labeled as follows: - Point \(A\): Located on the circumference of the circle. - Point \(B\): Located on the circumference of the circle. - Point \(C\): The vertex of the right angle, also located on the circumference of the circle. A right triangle is formed with the right angle at vertex \(C\), positioned at the top of the circle. #### Details - The circle is outlined with a green circumference. - The hypotenuse \( \overline{AB} \) lies along the diameter of the circle. - The length of the base \( \overline{AC} \) is given as 12 cm. - Line segments \( \overline{AC} \) and \( \overline{BC} \) form a right angle (90°) at \(C\). - A shaded purple region is created by the area of the circle outside the triangle and underneath the hypotenuse. ### Key Concepts 1. **Right Triangle Properties**: - In a right triangle inscribed within a circle, the hypotenuse is always the diameter of the circle. This is a consequence of the inscribed angle theorem, which states that an angle subtended by a diameter is a right angle. 2. **Circle Geometry**: - The relationship between the right triangle and the circle can be used to solve various geometric problems, such as finding the circle's radius, the length of the other sides of the triangle, and areas related to both the triangle and the circle. #### Example Problem: Given \( \overline{AC} = 12 \, \text{cm} \), and knowing \(\angle ACB = 90°\), find the radius of the circle. **Solution**: 1. Identify that \(\overline{AB}\) is the diameter of the circle. 2. Use the Pythagorean theorem to find \(\overline{BC}\) if needed. For right triangles inscribed in a circle, the hypotenuse \( \overline{AB} \) can be directly used to determine the diameter. This diagram is a useful educational tool for visualizing the relationship between right triangles and circles, illustrating key geometric principles, and solving related mathematical problems.
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