The sides of a rectangle are 15 cm and 17 cm. What is the measure of the angle formed by the long side and a diagonal of the rectangle, to the nearest degree? Show all work.
The sides of a rectangle are 15 cm and 17 cm. What is the measure of the angle formed by the long side and a diagonal of the rectangle, to the nearest degree? Show all work.
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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![**Geometry Problem: Calculating the Angle in a Rectangle**
**Problem Statement:**
The sides of a rectangle are 15 cm and 17 cm. What is the measure of the angle formed by the long side and a diagonal of the rectangle, to the nearest degree? Show all work.
**Solution:**
1. **Identify the components:**
- Length of the rectangle (long side), \( a = 17 \) cm
- Width of the rectangle (short side), \( b = 15 \) cm
2. **Calculate the length of the diagonal:**
Using the Pythagorean theorem:
\[ d = \sqrt{a^2 + b^2} \]
\[ d = \sqrt{17^2 + 15^2} \]
\[ d = \sqrt{289 + 225} \]
\[ d = \sqrt{514} \]
\[ d \approx 22.68 \text{ cm} \]
3. **Determine the angle:**
The angle \(\theta\) formed by the long side and the diagonal can be found using the cosine function:
\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \]
\[ \cos(\theta) = \frac{a}{d} \]
\[ \cos(\theta) = \frac{17}{22.68} \]
\[ \cos(\theta) \approx 0.7497 \]
4. **Calculate \(\theta\):**
\[ \theta = \cos^{-1}(0.7497) \]
\[ \theta \approx 41.41^\circ \]
Therefore, the measure of the angle formed by the long side and the diagonal of the rectangle is approximately 41 degrees to the nearest degree.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2f3e5dad-3eef-4af4-8432-92621aeb9e2a%2Ff4e3d99d-6a72-45f8-b542-9a7b51a7e483%2Fvnt2bm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Geometry Problem: Calculating the Angle in a Rectangle**
**Problem Statement:**
The sides of a rectangle are 15 cm and 17 cm. What is the measure of the angle formed by the long side and a diagonal of the rectangle, to the nearest degree? Show all work.
**Solution:**
1. **Identify the components:**
- Length of the rectangle (long side), \( a = 17 \) cm
- Width of the rectangle (short side), \( b = 15 \) cm
2. **Calculate the length of the diagonal:**
Using the Pythagorean theorem:
\[ d = \sqrt{a^2 + b^2} \]
\[ d = \sqrt{17^2 + 15^2} \]
\[ d = \sqrt{289 + 225} \]
\[ d = \sqrt{514} \]
\[ d \approx 22.68 \text{ cm} \]
3. **Determine the angle:**
The angle \(\theta\) formed by the long side and the diagonal can be found using the cosine function:
\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \]
\[ \cos(\theta) = \frac{a}{d} \]
\[ \cos(\theta) = \frac{17}{22.68} \]
\[ \cos(\theta) \approx 0.7497 \]
4. **Calculate \(\theta\):**
\[ \theta = \cos^{-1}(0.7497) \]
\[ \theta \approx 41.41^\circ \]
Therefore, the measure of the angle formed by the long side and the diagonal of the rectangle is approximately 41 degrees to the nearest degree.
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