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30. In circle P, find the length of BD if the length of diameter AD is 12 inches. B 100⁰ 10

30. In circle P, find the length of BD if the length of diameter AD is 12 inches. B 100⁰ 10

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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Question
### Problem 30: Circle Geometry #### Problem Statement: In circle \( P \), find the length of \( BD \) if the length of diameter \( AD \) is 12 inches. #### Given: - Diameter \( AD \) = 12 inches. - \( \angle APB = 100^\circ \). #### Diagram Explanation: The diagram shows a circle with center \( P \). The circle includes points \( A \), \( B \), \( C \), and \( D \) positioned on the circumference, with points \( A \) and \( D \) defining the diameter \( AD \). The diameter spans across the circle passing through its center \( P \). An angle \( \angle APB \) is formed inside the circle with a measure of \( 100^\circ \). The points associated with this angle are: - \( A \) positioned on the circle's circumference at one end of the diameter. - \( P \) being the center of the circle. - \( B \) positioned on the circle’s circumference not along the diameter. Additionally, there is a chord \( BD \) which needs to be determined. #### Solution Approach: To solve for \( BD \), we can use properties of the circle and the given angle: 1. \( AD \) is the diameter, therefore, \( \overline{AP} = \overline{PD} = 6 \) inches because the radius is half of the diameter. 2. Using the angle inscribed in the semicircle theorem: - The sum of angles around point \( P \) within the semicircle should be \( 180^\circ \). - Knowing \( \overline{APC} = 100^\circ \), the exterior angle \( \overline{APD}\) would be complementary inside the triangle. 3. Identify any necessary trigonometric properties or relationships to solve \( \overline{BD} \). Students would apply these concepts to find the required length, solving through geometric properties or trigonometric relationships based on the circle's properties. Understanding how to use angles, chords, and diameter in the circle will facilitate finding lengths and proving geometric theorems.
Transcribed Image Text:### Problem 30: Circle Geometry #### Problem Statement: In circle \( P \), find the length of \( BD \) if the length of diameter \( AD \) is 12 inches. #### Given: - Diameter \( AD \) = 12 inches. - \( \angle APB = 100^\circ \). #### Diagram Explanation: The diagram shows a circle with center \( P \). The circle includes points \( A \), \( B \), \( C \), and \( D \) positioned on the circumference, with points \( A \) and \( D \) defining the diameter \( AD \). The diameter spans across the circle passing through its center \( P \). An angle \( \angle APB \) is formed inside the circle with a measure of \( 100^\circ \). The points associated with this angle are: - \( A \) positioned on the circle's circumference at one end of the diameter. - \( P \) being the center of the circle. - \( B \) positioned on the circle’s circumference not along the diameter. Additionally, there is a chord \( BD \) which needs to be determined. #### Solution Approach: To solve for \( BD \), we can use properties of the circle and the given angle: 1. \( AD \) is the diameter, therefore, \( \overline{AP} = \overline{PD} = 6 \) inches because the radius is half of the diameter. 2. Using the angle inscribed in the semicircle theorem: - The sum of angles around point \( P \) within the semicircle should be \( 180^\circ \). - Knowing \( \overline{APC} = 100^\circ \), the exterior angle \( \overline{APD}\) would be complementary inside the triangle. 3. Identify any necessary trigonometric properties or relationships to solve \( \overline{BD} \). Students would apply these concepts to find the required length, solving through geometric properties or trigonometric relationships based on the circle's properties. Understanding how to use angles, chords, and diameter in the circle will facilitate finding lengths and proving geometric theorems.
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