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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what is the area of the shaded region below?
![**What is the area of the shaded region below?**
### Diagram Explanation:
The image displays a circle with a right-angled triangle inscribed within it. The vertices of the triangle are labeled A, B, and C. Here are specific details:
- Triangle \( \triangle ABC \) is a right-angled triangle with \( \angle ABC = 90^\circ \).
- The circle's radius is equal to the hypotenuse of \( \triangle ABC \).
- The length of side \( AB \) is given as 12 cm.
- Point A and point C lie on the circumference of the circle.
- The region shaded in purple is the area we need to find.
### Calculation Steps:
1. **Finding the Length of BC (using Pythagoras' Theorem)**:
\( AB \) is 12 cm. Let's denote the length of \( BC \) as \( a \) and the hypotenuse \( AC \) as \( c \).
Since \( \triangle ABC \) is right-angled at B:
\[
AC = c = \sqrt{AB^2 + BC^2}
\]
2. **Finding the Area of the Triangle**:
The area of \( \triangle ABC \) can be calculated using:
\[
\text{Area of } \triangle ABC = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times AB \times BC
\]
3. **Finding the Area of the Sector**:
The area of the sector with the angle \( \angle ACB \) can be calculated if we know the radius of the circle and the central angle (which would be 90 degrees since \( \triangle ABC \) is a right-angled triangle).
4. **Calculating the Shaded Region**:
The shaded region is part of the circle outside the triangle, so it can be found by subtracting the area of \( \triangle ABC \) from the area of the sector.
\[
\text{Area of Shaded Region} = \text{Area of Sector} - \text{Area of Triangle}
\]
With these steps, you'll be able to find the area of the shaded region in the given diagram.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0d1376b4-854f-4844-8630-994d3c61ce3e%2Fb1b1460c-abe4-4642-b230-d0f6436b3cf5%2F7s2ga2j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**What is the area of the shaded region below?**
### Diagram Explanation:
The image displays a circle with a right-angled triangle inscribed within it. The vertices of the triangle are labeled A, B, and C. Here are specific details:
- Triangle \( \triangle ABC \) is a right-angled triangle with \( \angle ABC = 90^\circ \).
- The circle's radius is equal to the hypotenuse of \( \triangle ABC \).
- The length of side \( AB \) is given as 12 cm.
- Point A and point C lie on the circumference of the circle.
- The region shaded in purple is the area we need to find.
### Calculation Steps:
1. **Finding the Length of BC (using Pythagoras' Theorem)**:
\( AB \) is 12 cm. Let's denote the length of \( BC \) as \( a \) and the hypotenuse \( AC \) as \( c \).
Since \( \triangle ABC \) is right-angled at B:
\[
AC = c = \sqrt{AB^2 + BC^2}
\]
2. **Finding the Area of the Triangle**:
The area of \( \triangle ABC \) can be calculated using:
\[
\text{Area of } \triangle ABC = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times AB \times BC
\]
3. **Finding the Area of the Sector**:
The area of the sector with the angle \( \angle ACB \) can be calculated if we know the radius of the circle and the central angle (which would be 90 degrees since \( \triangle ABC \) is a right-angled triangle).
4. **Calculating the Shaded Region**:
The shaded region is part of the circle outside the triangle, so it can be found by subtracting the area of \( \triangle ABC \) from the area of the sector.
\[
\text{Area of Shaded Region} = \text{Area of Sector} - \text{Area of Triangle}
\]
With these steps, you'll be able to find the area of the shaded region in the given diagram.
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