help me answer these questions please ### 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: \[ \frac{24}{14} = \frac{12}{7} \] -

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Answered: 11. The scale factor of Triangle A:B is…

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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Question

help me answer these questions please

### 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:
\[
\frac{24}{14} = \frac{12}{7}
\]
-

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