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
Related questions
Question
![### Problem Statement:
In the given right-angled triangle ΔABC, angle ∠C is 90°. The vertices are labeled as follows:
- A at the left-most point,
- B at the top-right point, and
- C at the bottom-right point where the right angle is formed.
The sides of the triangle are designated as:
- Side opposite to angle A (between B and C) is 'a',
- Side adjacent to angle A (between A and C) is 'b',
- Hypotenuse (between A and B) is 'c'.
Given:
- Length of the hypotenuse, c = 15 units
- Measure of angle ∠B = 78°
Question:
- Find the length of side 'a'.
### Diagram:
Here we have a right-angled triangle with:
- Right angle at C (∠C = 90°)
- Hypotenuse (c) connecting points A and B
- Vertical side (a) connecting points B and C
- Horizontal side (b) connecting points A and C.
### Solution:
To find the length of side 'a', we utilize trigonometric relationships in the right-angled triangle.
Given that:
- \( c = 15 \)
- \( \angle B = 78° \)
To find side 'a', observe that in triangle ΔABC:
\[
\sin(\angle B) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c}
\]
Thus,
\[
\sin(78°) = \frac{a}{15}
\]
We can solve for 'a':
\[
a = 15 \times \sin(78°)
\]
Using a calculator to find \( \sin(78°) \):
\[
\sin(78°) \approx 0.9781
\]
Therefore:
\[
a \approx 15 \times 0.9781 \approx 14.67
\]
So, the length of side 'a' is approximately 14.67 units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc797aff8-bf05-4e13-9797-dfba08dadef9%2Fde317083-0126-40ad-8460-d4d248a14cc1%2Fio4n4i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
In the given right-angled triangle ΔABC, angle ∠C is 90°. The vertices are labeled as follows:
- A at the left-most point,
- B at the top-right point, and
- C at the bottom-right point where the right angle is formed.
The sides of the triangle are designated as:
- Side opposite to angle A (between B and C) is 'a',
- Side adjacent to angle A (between A and C) is 'b',
- Hypotenuse (between A and B) is 'c'.
Given:
- Length of the hypotenuse, c = 15 units
- Measure of angle ∠B = 78°
Question:
- Find the length of side 'a'.
### Diagram:
Here we have a right-angled triangle with:
- Right angle at C (∠C = 90°)
- Hypotenuse (c) connecting points A and B
- Vertical side (a) connecting points B and C
- Horizontal side (b) connecting points A and C.
### Solution:
To find the length of side 'a', we utilize trigonometric relationships in the right-angled triangle.
Given that:
- \( c = 15 \)
- \( \angle B = 78° \)
To find side 'a', observe that in triangle ΔABC:
\[
\sin(\angle B) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c}
\]
Thus,
\[
\sin(78°) = \frac{a}{15}
\]
We can solve for 'a':
\[
a = 15 \times \sin(78°)
\]
Using a calculator to find \( \sin(78°) \):
\[
\sin(78°) \approx 0.9781
\]
Therefore:
\[
a \approx 15 \times 0.9781 \approx 14.67
\]
So, the length of side 'a' is approximately 14.67 units.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, geometry and related others by exploring similar questions and additional content below.Recommended textbooks for you
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning