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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![1. Find the area of the regular triangle with a side length of 10 cm.
The image shows an equilateral triangle (a triangle with all sides of equal length) with each side labeled as 10 cm.
To find the area of an equilateral triangle, you can use the formula:
\[ \text{Area} = \frac{\sqrt{3}}{4} s^2 \]
where \( s \) is the length of a side of the triangle.
In this case, \( s = 10 \) cm.
By substituting the value into the formula, we get:
\[ \text{Area} = \frac{\sqrt{3}}{4} \times 10^2 \]
\[ \text{Area} = \frac{\sqrt{3}}{4} \times 100 \]
\[ \text{Area} \approx 43.3 \, \text{cm}^2 \]
So, the area of the equilateral triangle with a side length of 10 cm is approximately 43.3 square centimeters.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F94e3dc3a-5138-4de9-a130-9782421a1710%2F7f06859f-7314-4c88-aaaf-98f8bea7effc%2Fw0ms6us_processed.png&w=3840&q=75)
Transcribed Image Text:1. Find the area of the regular triangle with a side length of 10 cm.
The image shows an equilateral triangle (a triangle with all sides of equal length) with each side labeled as 10 cm.
To find the area of an equilateral triangle, you can use the formula:
\[ \text{Area} = \frac{\sqrt{3}}{4} s^2 \]
where \( s \) is the length of a side of the triangle.
In this case, \( s = 10 \) cm.
By substituting the value into the formula, we get:
\[ \text{Area} = \frac{\sqrt{3}}{4} \times 10^2 \]
\[ \text{Area} = \frac{\sqrt{3}}{4} \times 100 \]
\[ \text{Area} \approx 43.3 \, \text{cm}^2 \]
So, the area of the equilateral triangle with a side length of 10 cm is approximately 43.3 square centimeters.
![### Problem 3: Solve for \( x \)
#### Description:
The given problem involves finding the value of \( x \) in a geometric figure. The figure provided is a polygon with five angles, one of which is a right angle. The specific measures of the five angles are as follows:
- A right angle, which measures \( 90^\circ \)
- One angle labeled \( 102^\circ \)
- One angle labeled \( 54^\circ \)
- Two angles labeled \( 3x^\circ \) each
#### Steps to Solve:
1. **Identify and Use the Sum of Interior Angles:**
The sum of the interior angles of a polygon can be calculated using the formula:
\[
(n-2) \times 180^\circ
\]
where \( n \) is the number of sides (or angles) in the polygon. Since the given figure is a pentagon (\( n = 5 \)), the sum of its interior angles is:
\[
(5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]
2. **Set Up the Equation:**
The sum of the given angles should equal \( 540^\circ \):
\[
90^\circ + 102^\circ + 54^\circ + 3x^\circ + 3x^\circ = 540^\circ
\]
Simplify the sum of the constant angles:
\[
246^\circ + 6x^\circ = 540^\circ
\]
3. **Isolate \( x \):**
Subtract \( 246^\circ \) from both sides of the equation:
\[
6x^\circ = 540^\circ - 246^\circ
\]
\[
6x^\circ = 294^\circ
\]
Divide both sides by 6:
\[
x = \frac{294^\circ}{6}
\]
\[
x = 49^\circ
\]
Thus, the solution is \( x = 49^\circ \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F94e3dc3a-5138-4de9-a130-9782421a1710%2F7f06859f-7314-4c88-aaaf-98f8bea7effc%2Fv404y8_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 3: Solve for \( x \)
#### Description:
The given problem involves finding the value of \( x \) in a geometric figure. The figure provided is a polygon with five angles, one of which is a right angle. The specific measures of the five angles are as follows:
- A right angle, which measures \( 90^\circ \)
- One angle labeled \( 102^\circ \)
- One angle labeled \( 54^\circ \)
- Two angles labeled \( 3x^\circ \) each
#### Steps to Solve:
1. **Identify and Use the Sum of Interior Angles:**
The sum of the interior angles of a polygon can be calculated using the formula:
\[
(n-2) \times 180^\circ
\]
where \( n \) is the number of sides (or angles) in the polygon. Since the given figure is a pentagon (\( n = 5 \)), the sum of its interior angles is:
\[
(5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
\]
2. **Set Up the Equation:**
The sum of the given angles should equal \( 540^\circ \):
\[
90^\circ + 102^\circ + 54^\circ + 3x^\circ + 3x^\circ = 540^\circ
\]
Simplify the sum of the constant angles:
\[
246^\circ + 6x^\circ = 540^\circ
\]
3. **Isolate \( x \):**
Subtract \( 246^\circ \) from both sides of the equation:
\[
6x^\circ = 540^\circ - 246^\circ
\]
\[
6x^\circ = 294^\circ
\]
Divide both sides by 6:
\[
x = \frac{294^\circ}{6}
\]
\[
x = 49^\circ
\]
Thus, the solution is \( x = 49^\circ \).
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