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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### Finding the Measure of Angle ∠B
To determine the measure of angle ∠B, we will use trigonometric principles. Below is a right triangle labeled with vertex points C, A, and B. The triangle has the following side lengths:
- \( AC = 24 \) units
- \( AB = 25 \) units (hypotenuse)
- \( BC = 7 \) units
Additionally, we know that angle \( ∠C = 90° \) (a right angle).
---
#### Image Description:
The diagram within the image shows a right triangle labeled as follows:
- \( C \) is the leftmost vertex at the right angle.
- \( B \) is the bottom vertex.
- \( A \) is the top right vertex.
The side \( AC \) is labeled with a length of 24 units, and \( BC \) is labeled with a length of 7 units. The hypotenuse \( AB \) is labeled with a length of 25 units.
---
#### Task:
**Find the measure of ∠B and round to the nearest whole number.**
#### Choices:
- \( \text{ } \) \( 16° \)
- \( \text{ } \) \( 17° \)
- \( \text{ } \) \( 44° \)
- \( \text{ } \) \( 46° \)
- \( \text{ } \) \( 73° \)
- \( \text{ } \) \( 74° \)
To find the measure of angle ∠B, we use the sine, cosine, or tangent ratio. Here, we will use the tangent ratio because we have the lengths of the opposite side \( (BC) \) and the adjacent side \( (AC) \):
\[ \text{tan(∠B)} = \frac{\text{opposite}}{\text{adjacent}} = \frac{BC}{AC} = \frac{7}{24} \]
We then take the inverse tangent (arctan) to find ∠B:
\[ ∠B = \text{tan}^{-1} \Big( \frac{7}{24} \Big) \approx 16.26° \]
Rounding to the nearest whole number:
\[ ∠B ≈ 16° \]
So, the correct answer is \( 16° \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7f3d50e5-a330-4a3e-9ccb-00c03d665fb0%2Fdfe8a2cb-8b22-4046-81ef-da881371ca03%2Foeoftap_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
### Finding the Measure of Angle ∠B
To determine the measure of angle ∠B, we will use trigonometric principles. Below is a right triangle labeled with vertex points C, A, and B. The triangle has the following side lengths:
- \( AC = 24 \) units
- \( AB = 25 \) units (hypotenuse)
- \( BC = 7 \) units
Additionally, we know that angle \( ∠C = 90° \) (a right angle).
---
#### Image Description:
The diagram within the image shows a right triangle labeled as follows:
- \( C \) is the leftmost vertex at the right angle.
- \( B \) is the bottom vertex.
- \( A \) is the top right vertex.
The side \( AC \) is labeled with a length of 24 units, and \( BC \) is labeled with a length of 7 units. The hypotenuse \( AB \) is labeled with a length of 25 units.
---
#### Task:
**Find the measure of ∠B and round to the nearest whole number.**
#### Choices:
- \( \text{ } \) \( 16° \)
- \( \text{ } \) \( 17° \)
- \( \text{ } \) \( 44° \)
- \( \text{ } \) \( 46° \)
- \( \text{ } \) \( 73° \)
- \( \text{ } \) \( 74° \)
To find the measure of angle ∠B, we use the sine, cosine, or tangent ratio. Here, we will use the tangent ratio because we have the lengths of the opposite side \( (BC) \) and the adjacent side \( (AC) \):
\[ \text{tan(∠B)} = \frac{\text{opposite}}{\text{adjacent}} = \frac{BC}{AC} = \frac{7}{24} \]
We then take the inverse tangent (arctan) to find ∠B:
\[ ∠B = \text{tan}^{-1} \Big( \frac{7}{24} \Big) \approx 16.26° \]
Rounding to the nearest whole number:
\[ ∠B ≈ 16° \]
So, the correct answer is \( 16° \).
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