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What is the area of the shaded region below?

What is the area of the shaded region below?

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: What is the area of the shaded region below?** [Image Explanation] The image displays a circle with a triangle inscribed within it. The circle is outlined in green. The triangle is outlined in blue and has one vertex at the center of the circle, labeled as point \( B \), and the other two vertices on the circumference of the circle, labeled as point \( A \) and point \( C \). The segment \( \overline{BA} \) is a radius of the circle and measures 12 cm. There is a right angle (\( 90^\circ \)) at point \( B \). The segment \( \overline{BC} \) is also a radius of the circle. The shaded region is the area outside the right triangle \( \triangle ABC \) but inside the circle, enclosed by \( \overline{AC} \) and the arc \( \overset{\frown}{AC} \). To find the area of the shaded region, follow these steps: 1. **Calculate the area of the triangle \( \triangle ABC \)**: - The base \( \overline{BA} = 12 \text{ cm} \) - The height \( \overline{BC} = 12 \text{ cm} \) - Area of rectangle = \( \frac{1}{2} \times \text{base} \times \text{height} \) - Area \( \triangle ABC = \frac{1}{2} \times 12 \text{ cm} \times 12 \text{ cm} = 72 \text{ cm}^2 \) 2. **Calculate the area of the sector \( \overset{\frown}{AC} \)**: - The sector represents a quarter of the circle since the angle at the center is \( 90^\circ \). - The radius \( r = 12 \text{ cm} \) - Area of the circle \( = \pi r^2 = \pi (12 \text{ cm})^2 = 144\pi \text{ cm}^2 \) - Area of the sector of a circle (quarter circle) will be \( \frac{1}{4} \times 144\pi \text{ cm}^2 = 36\pi \text{ cm}^2\) 3. **Calculate the area of
Transcribed Image Text:**Question: What is the area of the shaded region below?** [Image Explanation] The image displays a circle with a triangle inscribed within it. The circle is outlined in green. The triangle is outlined in blue and has one vertex at the center of the circle, labeled as point \( B \), and the other two vertices on the circumference of the circle, labeled as point \( A \) and point \( C \). The segment \( \overline{BA} \) is a radius of the circle and measures 12 cm. There is a right angle (\( 90^\circ \)) at point \( B \). The segment \( \overline{BC} \) is also a radius of the circle. The shaded region is the area outside the right triangle \( \triangle ABC \) but inside the circle, enclosed by \( \overline{AC} \) and the arc \( \overset{\frown}{AC} \). To find the area of the shaded region, follow these steps: 1. **Calculate the area of the triangle \( \triangle ABC \)**: - The base \( \overline{BA} = 12 \text{ cm} \) - The height \( \overline{BC} = 12 \text{ cm} \) - Area of rectangle = \( \frac{1}{2} \times \text{base} \times \text{height} \) - Area \( \triangle ABC = \frac{1}{2} \times 12 \text{ cm} \times 12 \text{ cm} = 72 \text{ cm}^2 \) 2. **Calculate the area of the sector \( \overset{\frown}{AC} \)**: - The sector represents a quarter of the circle since the angle at the center is \( 90^\circ \). - The radius \( r = 12 \text{ cm} \) - Area of the circle \( = \pi r^2 = \pi (12 \text{ cm})^2 = 144\pi \text{ cm}^2 \) - Area of the sector of a circle (quarter circle) will be \( \frac{1}{4} \times 144\pi \text{ cm}^2 = 36\pi \text{ cm}^2\) 3. **Calculate the area of
### Geometry and Circle Theorems: Problem Solving **Problem Statement:** The figure of circle A shown below has diameter PR which intersects QS at point B and the measurements shown. Calculate the following measures: - \( m \angle PSQ \) - \( m \angle QRS \) - \( m \overline{RS} \) - \( m \angle AQS \) - \( m \overline{PS} \) **Diagram and Details:** The diagram provided is a circle with center A. The circle includes the following elements: - The diameter PR, which has a midpoint at A. - The chords QS and PR, which intersect at point B within the circle. - \( \angle PAS = 35^\circ \) - \( \angle QAR = 130^\circ \) Points P, Q, R, and S are located on the circumference of the circle. Red lines connect these points forming the chords. ### Descriptive Analysis: 1. **Calculating \( m \angle PSQ \):** - Using the inscribed angle theorem, \( \angle PSQ \) subtends the same arc as \( \angle PAQ \). Therefore: \( m \angle PSQ = \frac{1}{2}m \angle PAQ = \frac{1}{2}(105^\circ) = 52.5^\circ \). 2. **Finding \( m \angle QRS \):** - This is an exterior angle, equivalent to the sum of opposite interior angles in the cyclic quadrilateral PSQR: \( m \angle QRS = 180^\circ - \angle QRB = 180^\circ - 50^\circ = 130^\circ \). 3. **Determining \( m \overline{RS} \):** - Use the information of other computed measures using geometric properties or trigonometric applications. 4. **Angle \( m \angle AQS \):** - Since \( A \) is the center and diameter equivalent \( m \angle AQS = 180^\circ - \angle 130^\circ = 50^\circ \) 5. **Calculating \( m \overline{PS}\):** - Similar approach follows their properties, which can be calculated with available measures from the circle. This problem entails understanding and applying circle theorems, recognizing geometric properties, and using critical problems-solving skills. Each
Transcribed Image Text:### Geometry and Circle Theorems: Problem Solving **Problem Statement:** The figure of circle A shown below has diameter PR which intersects QS at point B and the measurements shown. Calculate the following measures: - \( m \angle PSQ \) - \( m \angle QRS \) - \( m \overline{RS} \) - \( m \angle AQS \) - \( m \overline{PS} \) **Diagram and Details:** The diagram provided is a circle with center A. The circle includes the following elements: - The diameter PR, which has a midpoint at A. - The chords QS and PR, which intersect at point B within the circle. - \( \angle PAS = 35^\circ \) - \( \angle QAR = 130^\circ \) Points P, Q, R, and S are located on the circumference of the circle. Red lines connect these points forming the chords. ### Descriptive Analysis: 1. **Calculating \( m \angle PSQ \):** - Using the inscribed angle theorem, \( \angle PSQ \) subtends the same arc as \( \angle PAQ \). Therefore: \( m \angle PSQ = \frac{1}{2}m \angle PAQ = \frac{1}{2}(105^\circ) = 52.5^\circ \). 2. **Finding \( m \angle QRS \):** - This is an exterior angle, equivalent to the sum of opposite interior angles in the cyclic quadrilateral PSQR: \( m \angle QRS = 180^\circ - \angle QRB = 180^\circ - 50^\circ = 130^\circ \). 3. **Determining \( m \overline{RS} \):** - Use the information of other computed measures using geometric properties or trigonometric applications. 4. **Angle \( m \angle AQS \):** - Since \( A \) is the center and diameter equivalent \( m \angle AQS = 180^\circ - \angle 130^\circ = 50^\circ \) 5. **Calculating \( m \overline{PS}\):** - Similar approach follows their properties, which can be calculated with available measures from the circle. This problem entails understanding and applying circle theorems, recognizing geometric properties, and using critical problems-solving skills. Each
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