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10) Explain how you could go about finding the area of the green shaded region. You do not have to give a numerical answer (unless you want to.) 8 cm Your answer A 8 cm U

10) Explain how you could go about finding the area of the green shaded region. You do not have to give a numerical answer (unless you want to.) 8 cm Your answer A 8 cm U

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 10: **Explain how you could go about finding the area of the green shaded region. You do not have to give a numerical answer (unless you want to).** #### Diagram Description: The diagram provided is a circle with a radius of 8 cm. Inside the circle, there is a right-angled triangle \( \triangle ABC \) with: - Vertex \( A \) at the center of the circle. - Vertex \( B \) on the circumference of the circle. - Vertex \( C \) on the circumference of the circle. The right angle is at \( A \). The base \( AC \) and height \( AB \) of the triangle are each 8 cm. The green shaded region is the portion of the circle outside of the triangle but inside the sector \( BAC \). #### Steps to Find the Area of the Green Shaded Region: 1. **Calculate the Area of the Sector \( BAC \):** - Since \( \triangle ABC \) is a right-angled triangle at \( A \), the angle \( BAC \) is 90 degrees. - The area of a sector of a circle is given by \( \frac{\theta}{360^\circ} \times \pi r^2 \), where \( \theta \) is the central angle and \( r \) is the radius. - Substitute \( \theta = 90^\circ \) and \( r = 8 \) cm into the formula: \( \frac{90^\circ}{360^\circ} \times \pi \times (8 \, \text{cm})^2 \). 2. **Calculate the Area of \( \triangle ABC \):** - Use the formula for the area of a right-angled triangle: \( \frac{1}{2} \times \text{base} \times \text{height} \). - Substitute the base \( AC = 8 \, \text{cm} \) and height \( AB = 8 \, \text{cm} \): \( \frac{1}{2} \times 8 \, \text{cm} \times 8 \, \text{cm} \). 3. **Find the Area of the Green Shaded Region:** - Subtract the area of the triangle \( \triangle ABC \) from the area of the sector \( BAC \): \[ \text{Area of the Green
Transcribed Image Text:### Question 10: **Explain how you could go about finding the area of the green shaded region. You do not have to give a numerical answer (unless you want to).** #### Diagram Description: The diagram provided is a circle with a radius of 8 cm. Inside the circle, there is a right-angled triangle \( \triangle ABC \) with: - Vertex \( A \) at the center of the circle. - Vertex \( B \) on the circumference of the circle. - Vertex \( C \) on the circumference of the circle. The right angle is at \( A \). The base \( AC \) and height \( AB \) of the triangle are each 8 cm. The green shaded region is the portion of the circle outside of the triangle but inside the sector \( BAC \). #### Steps to Find the Area of the Green Shaded Region: 1. **Calculate the Area of the Sector \( BAC \):** - Since \( \triangle ABC \) is a right-angled triangle at \( A \), the angle \( BAC \) is 90 degrees. - The area of a sector of a circle is given by \( \frac{\theta}{360^\circ} \times \pi r^2 \), where \( \theta \) is the central angle and \( r \) is the radius. - Substitute \( \theta = 90^\circ \) and \( r = 8 \) cm into the formula: \( \frac{90^\circ}{360^\circ} \times \pi \times (8 \, \text{cm})^2 \). 2. **Calculate the Area of \( \triangle ABC \):** - Use the formula for the area of a right-angled triangle: \( \frac{1}{2} \times \text{base} \times \text{height} \). - Substitute the base \( AC = 8 \, \text{cm} \) and height \( AB = 8 \, \text{cm} \): \( \frac{1}{2} \times 8 \, \text{cm} \times 8 \, \text{cm} \). 3. **Find the Area of the Green Shaded Region:** - Subtract the area of the triangle \( \triangle ABC \) from the area of the sector \( BAC \): \[ \text{Area of the Green
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