2. Consider the figure, which consists of a triangle ABC and a sector of a circle with a central angle of ZCBD = 40°. %3D C 8. D A 7 9. В (a) Find the length of the arc CD. Round the answer to at least one decimal place. Answer: (b) Find the total area of the shape ABDC. Round the answer to at least one decimal place. Answer:
2. Consider the figure, which consists of a triangle ABC and a sector of a circle with a central angle of ZCBD = 40°. %3D C 8. D A 7 9. В (a) Find the length of the arc CD. Round the answer to at least one decimal place. Answer: (b) Find the total area of the shape ABDC. Round the answer to at least one decimal place. Answer:
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
Related questions
Question
![**Geometry Problem**
**2. Consider the figure, which consists of a triangle \( ABC \) and a sector of a circle with a central angle of \( \angle CBD = 40^\circ \).**
![Diagram consisting of a triangle \(ABC\) and a sector with a central angle of \(40^\circ\). The lengths of \(AB\) and \(AC\) are labeled as \(9\) and \(8\) respectively. The point \(D\) is on the arc. The length \(BC\) is labeled as \(7\).]
### (a) Find the length of the arc \( \overset{\frown}{CD} \). Round the answer to at least one decimal place.
**Answer:** [ ]
### (b) Find the total area of the shape \(ABDC\). Round the answer to at least one decimal place.
**Answer:** [ ]
---
To solve the problems, apply the following concepts.
### (a) Length of Arc \( \overset{\frown}{CD} \)
To find the length of the arc \( \overset{\frown}{CD} \), use the formula for the length of an arc in a circle:
\[ L = r \theta \]
where:
- \( r \) is the radius of the circle.
- \( \theta \) is the central angle in radians.
First, convert \( \theta \) from degrees to radians:
\[ \theta = 40^\circ \times \frac{\pi}{180^\circ} \]
Then, using the given length \( BC = 7 \) as the radius \( r \):
\[ L = 7 \times (\theta \text{ in radians}) \]
### (b) Total Area of Shape \(ABDC\)
The total area of shape \(ABDC\) is the sum of the area of triangle \(ABC\) and the area of the sector \(CD\).
**1. Area of Triangle \(ABC\):**
Use the formula for the area of a triangle with sides \(a\), \(b\), and the included angle \(C\):
\[ \text{Area} = \frac{1}{2} ab \sin(C) \]
Here \(a = 8\), \(b = 9\), and \(C = 180^\circ - \angle CBD\). Use the given angle \(\angle CBD =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc20a2ae-86b1-4d65-9ff5-dee17119e71a%2F648502f9-47d9-4df9-8799-d830e78ed47b%2Fuqa5gr.png&w=3840&q=75)
Transcribed Image Text:**Geometry Problem**
**2. Consider the figure, which consists of a triangle \( ABC \) and a sector of a circle with a central angle of \( \angle CBD = 40^\circ \).**
![Diagram consisting of a triangle \(ABC\) and a sector with a central angle of \(40^\circ\). The lengths of \(AB\) and \(AC\) are labeled as \(9\) and \(8\) respectively. The point \(D\) is on the arc. The length \(BC\) is labeled as \(7\).]
### (a) Find the length of the arc \( \overset{\frown}{CD} \). Round the answer to at least one decimal place.
**Answer:** [ ]
### (b) Find the total area of the shape \(ABDC\). Round the answer to at least one decimal place.
**Answer:** [ ]
---
To solve the problems, apply the following concepts.
### (a) Length of Arc \( \overset{\frown}{CD} \)
To find the length of the arc \( \overset{\frown}{CD} \), use the formula for the length of an arc in a circle:
\[ L = r \theta \]
where:
- \( r \) is the radius of the circle.
- \( \theta \) is the central angle in radians.
First, convert \( \theta \) from degrees to radians:
\[ \theta = 40^\circ \times \frac{\pi}{180^\circ} \]
Then, using the given length \( BC = 7 \) as the radius \( r \):
\[ L = 7 \times (\theta \text{ in radians}) \]
### (b) Total Area of Shape \(ABDC\)
The total area of shape \(ABDC\) is the sum of the area of triangle \(ABC\) and the area of the sector \(CD\).
**1. Area of Triangle \(ABC\):**
Use the formula for the area of a triangle with sides \(a\), \(b\), and the included angle \(C\):
\[ \text{Area} = \frac{1}{2} ab \sin(C) \]
Here \(a = 8\), \(b = 9\), and \(C = 180^\circ - \angle CBD\). Use the given angle \(\angle CBD =
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