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Find arc length in terms of n and to the hundredths place. Find degree measurements to the nearest degree. Find all other measures to the nearest hundredths place. Find the length of AB. 1. A 2. 3. 7 in. 6 ft 120° 45° B 12 cm B.

Find arc length in terms of n and to the hundredths place. Find degree measurements to the nearest degree. Find all other measures to the nearest hundredths place. Find the length of AB. 1. A 2. 3. 7 in. 6 ft 120° 45° B 12 cm B.

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: Finding Arc Lengths and Measurements In this lesson, we will learn how to find the arc length in terms of π and to the hundredths place. We will also determine degree measurements to the nearest degree, and all other measures to the nearest hundredths place. Below are diagrams of three different problems to practice finding the length of arc \( \overset{\frown}{AB} \). #### Problem 1 - **Diagram Description**: A circle with center \( Q \). The radius from \( Q \) to \( A \) measures 7 inches. There is a right angle (90°) formed at \( Q \). #### Problem 2 - **Diagram Description**: A circle with center \( Q \). The radius from \( Q \) to \( A \) measures 6 feet. The central angle \( \angle AQB \) measures 120°. #### Problem 3 - **Diagram Description**: A circle with center \( Q \). The radius from \( Q \) to \( A \) measures 12 centimeters. The central angle \( \angle AQB \) measures 45°. ### Steps to Find Arc Length: 1. **Degree to Radian Conversion**: - The formula to convert degrees to radians: \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \) 2. **Finding Arc Length**: - The arc length \( s \) can be found using the formula: \( s = \theta \times r \), where \( \theta \) is in radians and \( r \) is the radius of the circle. #### Applying the Steps: - **Problem 1**: - \( \theta = 90° \times \frac{\pi}{180} = \frac{\pi}{2} \) - Arc Length \( \overset{\frown}{AB} = \frac{\pi}{2} \times 7 = 3.5\pi \) inches. - **Problem 2**: - \( \theta = 120° \times \frac{\pi}{180} = \frac{2\pi}{3} \) - Arc Length \( \overset{\frown}{AB} = \frac{2\pi}{3} \times 6 = 4\pi \) feet. - **Problem 3**: -
Transcribed Image Text:### Geometry: Finding Arc Lengths and Measurements In this lesson, we will learn how to find the arc length in terms of π and to the hundredths place. We will also determine degree measurements to the nearest degree, and all other measures to the nearest hundredths place. Below are diagrams of three different problems to practice finding the length of arc \( \overset{\frown}{AB} \). #### Problem 1 - **Diagram Description**: A circle with center \( Q \). The radius from \( Q \) to \( A \) measures 7 inches. There is a right angle (90°) formed at \( Q \). #### Problem 2 - **Diagram Description**: A circle with center \( Q \). The radius from \( Q \) to \( A \) measures 6 feet. The central angle \( \angle AQB \) measures 120°. #### Problem 3 - **Diagram Description**: A circle with center \( Q \). The radius from \( Q \) to \( A \) measures 12 centimeters. The central angle \( \angle AQB \) measures 45°. ### Steps to Find Arc Length: 1. **Degree to Radian Conversion**: - The formula to convert degrees to radians: \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \) 2. **Finding Arc Length**: - The arc length \( s \) can be found using the formula: \( s = \theta \times r \), where \( \theta \) is in radians and \( r \) is the radius of the circle. #### Applying the Steps: - **Problem 1**: - \( \theta = 90° \times \frac{\pi}{180} = \frac{\pi}{2} \) - Arc Length \( \overset{\frown}{AB} = \frac{\pi}{2} \times 7 = 3.5\pi \) inches. - **Problem 2**: - \( \theta = 120° \times \frac{\pi}{180} = \frac{2\pi}{3} \) - Arc Length \( \overset{\frown}{AB} = \frac{2\pi}{3} \times 6 = 4\pi \) feet. - **Problem 3**: -
### Circle Geometry: Finding Measures In circle \( \bigodot D \) shown below, \( \angle ADC \cong \angle BDC \). Find the indicated measure. 1. **\( m\angle ACB \)** 2. **\( m\angle CB\)** 3. **Length of \( \overset\frown{ACB} \)** 4. **Length of \( \overset\frown{CB} \)** 5. **\( m\overset\frown{ABC} \)** 6. **Length of \( \overset\frown{BAC} \)** ### Diagram Analysis: The diagram on the right consists of a circle with center \(D\). Inside the circle, there are points labeled as \(A\), \(B\), \(C\), and \(D\), where \(AB\), \(BC\), and \(CD\) are chord segments. Point \( D \) is also the vertex of two intersecting angles inside the circle. - \( \angle ADC = 40^\circ\) - The radius \( \overline{AD} = 4 \) inches. This configuration is used to find the specific measures indicated by the problems listed. ### Instructions to Students: To solve for the measures and lengths indicated, you should use principles of circle geometry, such as: - The measure of an inscribed angle is half the measure of the intercepted arc. - The lengths of arcs can be found using the formula \( \text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r \), where \( \theta \) is the angle in degrees and \( r \) is the radius of the circle. Please apply these concepts to find the specific values required in each part.
Transcribed Image Text:### Circle Geometry: Finding Measures In circle \( \bigodot D \) shown below, \( \angle ADC \cong \angle BDC \). Find the indicated measure. 1. **\( m\angle ACB \)** 2. **\( m\angle CB\)** 3. **Length of \( \overset\frown{ACB} \)** 4. **Length of \( \overset\frown{CB} \)** 5. **\( m\overset\frown{ABC} \)** 6. **Length of \( \overset\frown{BAC} \)** ### Diagram Analysis: The diagram on the right consists of a circle with center \(D\). Inside the circle, there are points labeled as \(A\), \(B\), \(C\), and \(D\), where \(AB\), \(BC\), and \(CD\) are chord segments. Point \( D \) is also the vertex of two intersecting angles inside the circle. - \( \angle ADC = 40^\circ\) - The radius \( \overline{AD} = 4 \) inches. This configuration is used to find the specific measures indicated by the problems listed. ### Instructions to Students: To solve for the measures and lengths indicated, you should use principles of circle geometry, such as: - The measure of an inscribed angle is half the measure of the intercepted arc. - The lengths of arcs can be found using the formula \( \text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r \), where \( \theta \) is the angle in degrees and \( r \) is the radius of the circle. Please apply these concepts to find the specific values required in each part.
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