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DO What is the mBC? E 26°

DO What is the mBC? E 26°

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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On the provided diagram, we have a circle with several key points and lines marked. Here's a breakdown: - The circle's center is labeled as point \( E \). - Lines drawn form a right angle at the center of the circle. - Point \( A \) is on the topmost part of the circle. - Point \( C \) is directly at the bottom of the circle. - Points \( B \) and \( D \) lie on the right and left sides of the circle respectively, forming a horizontal diameter. - Outside the circle and extending horizontally, point \( F \) lies on a line extending through point \( D \). Going into further detail: - Line \( AE \) extends vertically from the center to point \( A \). - The angle \( \angle AEB \) is marked as \( 26^\circ \). The primary question attached to this diagram is: "What is the measure of \( \angle BCF \)?" To solve this, we can use the following geometric principles and facts: 1. **Central Angles and Arc Measures**: Since \( \angle AEB \) has been given as \( 26^\circ \), and it is a central angle, it implies that arc \( AB \) equals \( 26^\circ \). 2. **Inscribed Angle Theorem**: The inscribed angle \( \angle BCF \) that subtends the same arc \( AB \) will measure half of the central angle. Therefore, \( \angle BCF = \frac{26^\circ}{2} = 13^\circ \). Answer: The measure of \( \angle BCF \) is \( 13^\circ \).
Transcribed Image Text:On the provided diagram, we have a circle with several key points and lines marked. Here's a breakdown: - The circle's center is labeled as point \( E \). - Lines drawn form a right angle at the center of the circle. - Point \( A \) is on the topmost part of the circle. - Point \( C \) is directly at the bottom of the circle. - Points \( B \) and \( D \) lie on the right and left sides of the circle respectively, forming a horizontal diameter. - Outside the circle and extending horizontally, point \( F \) lies on a line extending through point \( D \). Going into further detail: - Line \( AE \) extends vertically from the center to point \( A \). - The angle \( \angle AEB \) is marked as \( 26^\circ \). The primary question attached to this diagram is: "What is the measure of \( \angle BCF \)?" To solve this, we can use the following geometric principles and facts: 1. **Central Angles and Arc Measures**: Since \( \angle AEB \) has been given as \( 26^\circ \), and it is a central angle, it implies that arc \( AB \) equals \( 26^\circ \). 2. **Inscribed Angle Theorem**: The inscribed angle \( \angle BCF \) that subtends the same arc \( AB \) will measure half of the central angle. Therefore, \( \angle BCF = \frac{26^\circ}{2} = 13^\circ \). Answer: The measure of \( \angle BCF \) is \( 13^\circ \).
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