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Find the indicated measure. 10. Length of AB 11. 135° d= 20 cm

Find the indicated measure. 10. Length of AB 11. 135° d= 20 cm

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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### 13. WRITING: **Describe the difference between an arc measure and an arc length.** An arc measure is the degree measure of the central angle that intercepts the arc, typically expressed in degrees or radians. For example, if an arc is intercepted by a 60° angle, its arc measure is 60°. On the other hand, arc length is the actual distance along the curved line forming the arc. It can be calculated using the formula: \[ \text{Arc Length} = \left(\frac{\text{Arc Measure}}{360}\right) \times 2\pi r \] ### 14. ERROR ANALYSIS: **Describe and correct the error in finding the length of GH.** #### Given Diagram: A circle with center \(C\) and radius \(5 \, \text{cm}\). The arc \(GH\) is intercepted by a 75° central angle. #### Calculation Shown: 1. \( \text{Arc length of } GH = m\overarc{GH} \cdot 2\pi \) 2. \( \text{Arc length of } GH = 75 \cdot 2\pi(5) \) 3. \( \text{Arc length of } GH = 75 \cdot 2\pi(5) \) 4. \( \text{Arc length of } GH = 750\pi \, \text{cm} \) #### Error Explanation: The error in the given calculation lies in the incorrect application of the arc length formula. The arc length should properly account for the proportion of the circle represented by the central angle. #### Correct Calculation: 1. Convert the given central angle from degrees to radians or use it directly in proportion to the full circle. 2. For degrees: \[ \text{Arc Length} = \left(\frac{\text{Arc Measure}}{360}\right) \times 2\pi r \] 3. Plug in the values: \[ \text{Arc Length} = \left(\frac{75}{360}\right) \times 2\pi(5) \] \[ \text{Arc Length} = \left(\frac{75}{360}\right) \times 10\pi \] \[ \text{Arc Length} = \left(\frac{75}{36}\right) \pi \] \[ \text{Arc Length} = 2.0833
Transcribed Image Text:### 13. WRITING: **Describe the difference between an arc measure and an arc length.** An arc measure is the degree measure of the central angle that intercepts the arc, typically expressed in degrees or radians. For example, if an arc is intercepted by a 60° angle, its arc measure is 60°. On the other hand, arc length is the actual distance along the curved line forming the arc. It can be calculated using the formula: \[ \text{Arc Length} = \left(\frac{\text{Arc Measure}}{360}\right) \times 2\pi r \] ### 14. ERROR ANALYSIS: **Describe and correct the error in finding the length of GH.** #### Given Diagram: A circle with center \(C\) and radius \(5 \, \text{cm}\). The arc \(GH\) is intercepted by a 75° central angle. #### Calculation Shown: 1. \( \text{Arc length of } GH = m\overarc{GH} \cdot 2\pi \) 2. \( \text{Arc length of } GH = 75 \cdot 2\pi(5) \) 3. \( \text{Arc length of } GH = 75 \cdot 2\pi(5) \) 4. \( \text{Arc length of } GH = 750\pi \, \text{cm} \) #### Error Explanation: The error in the given calculation lies in the incorrect application of the arc length formula. The arc length should properly account for the proportion of the circle represented by the central angle. #### Correct Calculation: 1. Convert the given central angle from degrees to radians or use it directly in proportion to the full circle. 2. For degrees: \[ \text{Arc Length} = \left(\frac{\text{Arc Measure}}{360}\right) \times 2\pi r \] 3. Plug in the values: \[ \text{Arc Length} = \left(\frac{75}{360}\right) \times 2\pi(5) \] \[ \text{Arc Length} = \left(\frac{75}{360}\right) \times 10\pi \] \[ \text{Arc Length} = \left(\frac{75}{36}\right) \pi \] \[ \text{Arc Length} = 2.0833
## Find the Indicated Measure ### 10. Length of \( \overline{AB} \) A figure of a circle is shown with a diameter \( \overline{AB} \) labeled as \( d = 20 \) cm. Point \( Q \) is the center of the circle, and \( \angle AQB \) is given as \( 135^\circ \). ### 11. Circumference of \( \bigodot Q \) A second figure of a circle is displayed with a radius \( \overline{OQ} \). Point \( Q \) is the center, \( \overline{AB} \) is the radius of the circle and it is given as \( 15 \) in. The angle at the center \( \angle AQB \) is \( 60^\circ \). ### 12. Radius of \( \bigodot Q \) A third figure of a circle is displayed with a radius \( \overline{QO} \). Point \( Q \) is the center, and it is labeled with a right angle at point A. The arc from center Q to point A to point B is \( 18\pi \) in. Each diagram provides specific measurements and angles to find the required parameters such as the length of a line segment, circumference, and the radius of the circles.
Transcribed Image Text:## Find the Indicated Measure ### 10. Length of \( \overline{AB} \) A figure of a circle is shown with a diameter \( \overline{AB} \) labeled as \( d = 20 \) cm. Point \( Q \) is the center of the circle, and \( \angle AQB \) is given as \( 135^\circ \). ### 11. Circumference of \( \bigodot Q \) A second figure of a circle is displayed with a radius \( \overline{OQ} \). Point \( Q \) is the center, \( \overline{AB} \) is the radius of the circle and it is given as \( 15 \) in. The angle at the center \( \angle AQB \) is \( 60^\circ \). ### 12. Radius of \( \bigodot Q \) A third figure of a circle is displayed with a radius \( \overline{QO} \). Point \( Q \) is the center, and it is labeled with a right angle at point A. The arc from center Q to point A to point B is \( 18\pi \) in. Each diagram provides specific measurements and angles to find the required parameters such as the length of a line segment, circumference, and the radius of the circles.
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