11. The scale factor of Triangle A:B is 2/3. A. В. Perimeter of Triangle B is 39 Find the Perimeter of Triangle B Area of Triangle A is 60. Find the Area of Triangle B
11. The scale factor of Triangle A:B is 2/3. A. В. Perimeter of Triangle B is 39 Find the Perimeter of Triangle B Area of Triangle A is 60. Find the Area of Triangle B
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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![### Similar Triangles and Scale Factors
### Problem 1:
**Given:**
- The scale factor of Triangle A:B is 2/3.
- Perimeter of Triangle B is 39.
- Area of Triangle A is 60.
**Questions:**
1. Find the Perimeter of Triangle A.
2. Find the Area of Triangle B.
### Solution:
1. **Finding the Perimeter of Triangle A:**
Using the scale factor of 2/3, we can set up the relationship between the perimeters of the two triangles:
\[
\frac{\text{Perimeter of Triangle A}}{\text{Perimeter of Triangle B}} = \frac{2}{3}
\]
Given:
\[
\text{Perimeter of Triangle B} = 39
\]
\[
\text{Perimeter of Triangle A} = \frac{2}{3} \times 39 = 26
\]
2. **Finding the Area of Triangle B:**
The scale factor affects the area by the square of the scale factor. Hence,
\[
\frac{\text{Area of Triangle A}}{\text{Area of Triangle B}} = \left(\frac{2}{3}\right)^2 = \frac{4}{9}
\]
Given:
\[
\text{Area of Triangle A} = 60
\]
\[
\text{Area of Triangle B} = \frac{4}{9} \times 60 = \frac{240}{9} = 26.\overline{66}
\]
---
### Problem 2:
**Given:**
- Triangle \( \Delta ABC \) has side lengths 42, 21, and 35 units.
- The shortest side of a triangle similar to \( \Delta ABC \) is 9 units long.
**Questions:**
1. Find the perimeter of the smaller triangle.
2. Find the area of the smaller triangle.
### Solution:
1. **Finding the Perimeter of the Smaller Triangle:**
The scale factor can be determined by comparing the shortest sides of the triangles:
\[
\text{Scale Factor} = \frac{9}{21} = \frac{3}{7}
\]
Using this scale factor to](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdadbed77-0193-4e9d-88df-390defe35d5d%2F27d79a8a-8372-44e0-bd49-ab38488d5205%2Fj7x5ux7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Similar Triangles and Scale Factors
### Problem 1:
**Given:**
- The scale factor of Triangle A:B is 2/3.
- Perimeter of Triangle B is 39.
- Area of Triangle A is 60.
**Questions:**
1. Find the Perimeter of Triangle A.
2. Find the Area of Triangle B.
### Solution:
1. **Finding the Perimeter of Triangle A:**
Using the scale factor of 2/3, we can set up the relationship between the perimeters of the two triangles:
\[
\frac{\text{Perimeter of Triangle A}}{\text{Perimeter of Triangle B}} = \frac{2}{3}
\]
Given:
\[
\text{Perimeter of Triangle B} = 39
\]
\[
\text{Perimeter of Triangle A} = \frac{2}{3} \times 39 = 26
\]
2. **Finding the Area of Triangle B:**
The scale factor affects the area by the square of the scale factor. Hence,
\[
\frac{\text{Area of Triangle A}}{\text{Area of Triangle B}} = \left(\frac{2}{3}\right)^2 = \frac{4}{9}
\]
Given:
\[
\text{Area of Triangle A} = 60
\]
\[
\text{Area of Triangle B} = \frac{4}{9} \times 60 = \frac{240}{9} = 26.\overline{66}
\]
---
### Problem 2:
**Given:**
- Triangle \( \Delta ABC \) has side lengths 42, 21, and 35 units.
- The shortest side of a triangle similar to \( \Delta ABC \) is 9 units long.
**Questions:**
1. Find the perimeter of the smaller triangle.
2. Find the area of the smaller triangle.
### Solution:
1. **Finding the Perimeter of the Smaller Triangle:**
The scale factor can be determined by comparing the shortest sides of the triangles:
\[
\text{Scale Factor} = \frac{9}{21} = \frac{3}{7}
\]
Using this scale factor to
![### Problem 12: Triangle Similarity
**Given:** Use the figure to find the values of \( x \), \( y \), and \( z \) that make \(\Delta DEF \sim \Delta HGF\).
**Figure Description:**
The figure contains two triangles, \(\Delta DEF\) and \(\Delta HGF\), arranged in such a way that they appear to be similar by the given information.
- In \(\Delta DEF\), the side \( DE \) measures 25 units, \( DF \) measures 24 units, and the angle \( \angle DEF = 16^\circ \).
- In \(\Delta HGF\), the side \( FG \) measures 14 units, \( GH \) measures \( 3y \) units, and \( HF \) measures \( 6z + 8 \) units. The angle \( \angle HGF \) is given as \( 2(x - 4)^\circ \).
Additionally:
- \(\angle EFD\) in \(\Delta DEF\) is \( x - 5^\circ \)
- \(\Delta HGF\) has an angle \( \angle HGF = 2(x - 4)^\circ\)
---
To solve this problem, use the properties of similar triangles, specifically that corresponding angles are equal and corresponding sides are proportional.
Steps to find \( x \), \( y \), and \( z \):
1. **Angle Correspondence:**
- Since \(\Delta DEF \sim \Delta HGF\), corresponding angles must be equal, leading to:
\[
\angle DEF = 16^\circ = \angle HGF
\]
Hence,
\(2(x - 4) = 16\%\):
\[
2(x - 4) = 16 \\
x - 4 = 8 \\
x = 12
\]
2. **Side Proportionality:**
- \(\frac{DF}{FG} = \frac{EF}{HF} = \frac{ED}{GH}\)
- Since \(DF = 24\) and \(FG = 14\), the ratio \(\frac{DF}{FG}\) is:
\[
\frac{24}{14} = \frac{12}{7}
\]
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdadbed77-0193-4e9d-88df-390defe35d5d%2F27d79a8a-8372-44e0-bd49-ab38488d5205%2Fnj0odge_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 12: Triangle Similarity
**Given:** Use the figure to find the values of \( x \), \( y \), and \( z \) that make \(\Delta DEF \sim \Delta HGF\).
**Figure Description:**
The figure contains two triangles, \(\Delta DEF\) and \(\Delta HGF\), arranged in such a way that they appear to be similar by the given information.
- In \(\Delta DEF\), the side \( DE \) measures 25 units, \( DF \) measures 24 units, and the angle \( \angle DEF = 16^\circ \).
- In \(\Delta HGF\), the side \( FG \) measures 14 units, \( GH \) measures \( 3y \) units, and \( HF \) measures \( 6z + 8 \) units. The angle \( \angle HGF \) is given as \( 2(x - 4)^\circ \).
Additionally:
- \(\angle EFD\) in \(\Delta DEF\) is \( x - 5^\circ \)
- \(\Delta HGF\) has an angle \( \angle HGF = 2(x - 4)^\circ\)
---
To solve this problem, use the properties of similar triangles, specifically that corresponding angles are equal and corresponding sides are proportional.
Steps to find \( x \), \( y \), and \( z \):
1. **Angle Correspondence:**
- Since \(\Delta DEF \sim \Delta HGF\), corresponding angles must be equal, leading to:
\[
\angle DEF = 16^\circ = \angle HGF
\]
Hence,
\(2(x - 4) = 16\%\):
\[
2(x - 4) = 16 \\
x - 4 = 8 \\
x = 12
\]
2. **Side Proportionality:**
- \(\frac{DF}{FG} = \frac{EF}{HF} = \frac{ED}{GH}\)
- Since \(DF = 24\) and \(FG = 14\), the ratio \(\frac{DF}{FG}\) is:
\[
\frac{24}{14} = \frac{12}{7}
\]
-
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