please help me answer questions 11 and 12 ### Problem 11**Given:**- Triangle $ \Delta ABC $ has side lengths 42 units, 21 units, and 35 units.- A triangle similar to $ \Delta ABC $ has its shortest side measuring 9 units.**Objective:**Find the perimeter and area of the smaller triangle.**Solution Steps:**1. **Identify the shortest side in $ \Delta ABC $:** - The side lengths given are 42 units, 21 units, and 35 units. - The shortest side is 21 units.2. **Determine the similarity ratio:** - The ratio of the corresponding sides of the similar triangles is determined by their shortest sides. - The shortest side of the similar triangle is 9 units, and in $ \Delta ABC $, it is 21 units. - Therefore, the similarity ratio (scaling factor) is $ \frac{9}{21} = \frac{3}{7} $.3. **Calculate the sides of the smaller triangle:** - Since the triangles are similar, all sides of the smaller triangle are scaled by the ratio $ \frac{3}{7} $. - Side lengths of the smaller triangle: - $ \frac{3}{7} \times 21 = 9 $ units - $ \frac{3}{7} \times 42 = 18 $ units - $ \frac{3}{7} \times 35 = 15 $ units4. **Find the perimeter of the smaller triangle:** - Perimeter $ P = 9 + 18 + 15 = 42 $ units.5. **Calculate the area of the smaller triangle:** - Use Heron’s formula for the area of a triangle with sides $ a = 9 $, $ b = 18 $, and $ c = 15 $. - First, calculate the semi-perimeter $ s $: $ s = \frac{a + b + c}{2} = \frac{9 + 18 + 15}{2} = 21 $ units. - Substitute into Heron’s formula: \[ \text{Area} = \sqrt{s \cdot (s - a) \cdot (s - b) \cdot (s - c)} = \sqrt{21 ### Problem Statement**Question 12:** Use the figure to find the values of $x$, $y$, and $z$ that make $\triangle DEF \sim \triangle HGF$.### Explanation of the DiagramThe diagram consists of two triangles: $\triangle DEF$ and $\triangle HGF$.- **Triangle DEF:** - $ \angle DEF = 16^\circ $ - $ DE = 25 $ - $ DF = 24 $ - $ EF = x - 5 $- **Triangle HGF:** - $ \angle HGF = 2(x - 4)^\circ $ - $ GF = 14 $ - $ HF = 6z + 8 $ - $ GH = 3y $Triangles $\triangle DEF$ and $\triangle HGF$ are given as similar ($\triangle DEF \sim \triangle HGF$), meaning corresponding angles are equal and corresponding sides are proportional.Since the triangles are similar:1. The corresponding angles are equal.2. The ratios of the lengths of corresponding sides are equal.### Steps to Find the Values of $x$, $y$, and $z$:1. **Angle Correlations:** - Given $\angle DEF = \angle HGF$, so: \[ 16^\circ = 2(x - 4)^\circ \] Solving for $x$: \[ 16 = 2(x - 4) \] \[ 16 = 2x - 8 \] \[ 2x = 24 \] \[ x = 12 \]2. **Side Ratios:** Since $\triangle DEF \sim \triangle HGF$: \[ \frac{DE}{GH} = \frac{DF}{GF} = \frac{EF}{HF} \] Using the sides: \[ \frac{25}{3y} = \frac{24}{14} = \frac{x - 5}{6z + 8} \] Simplifying, we get: - For $\frac{25}{3y} = \frac{24}{14}$: \[ 25 \cdot 14 = 24 \cd

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We need to use similar triangles formulaes to solve this question.

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2. Use the figure to find the values of x, y, and z that makes ADEF ~ AHGF. 25 16 X-5 F Зу H. 14 2(x-4)° 6z + 8 24

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ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

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please help me answer questions 11 and 12

### Problem 11

**Given:**
- Triangle $ \Delta ABC $ has side lengths 42 units, 21 units, and 35 units.
- A triangle similar to $ \Delta ABC $ has its shortest side measuring 9 units.

**Objective:**
Find the perimeter and area of the smaller triangle.

**Solution Steps:**

1. **Identify the shortest side in $ \Delta ABC $:**
- The side lengths given are 42 units, 21 units, and 35 units.
- The shortest side is 21 units.

2. **Determine the similarity ratio:**
- The ratio of the corresponding sides of the similar triangles is determined by their shortest sides.
- The shortest side of the similar triangle is 9 units, and in $ \Delta ABC $, it is 21 units.
- Therefore, the similarity ratio (scaling factor) is $ \frac{9}{21} = \frac{3}{7} $.

3. **Calculate the sides of the smaller triangle:**
- Since the triangles are similar, all sides of the smaller triangle are scaled by the ratio $ \frac{3}{7} $.
- Side lengths of the smaller triangle:
- $ \frac{3}{7} \times 21 = 9 $ units
- $ \frac{3}{7} \times 42 = 18 $ units
- $ \frac{3}{7} \times 35 = 15 $ units

4. **Find the perimeter of the smaller triangle:**
- Perimeter $ P = 9 + 18 + 15 = 42 $ units.

5. **Calculate the area of the smaller triangle:**
- Use Heron’s formula for the area of a triangle with sides $ a = 9 $, $ b = 18 $, and $ c = 15 $.
- First, calculate the semi-perimeter $ s $:
$ s = \frac{a + b + c}{2} = \frac{9 + 18 + 15}{2} = 21 $ units.
- Substitute into Heron’s formula:
\[
\text{Area} = \sqrt{s \cdot (s - a) \cdot (s - b) \cdot (s - c)} = \sqrt{21

### Problem Statement

**Question 12:**

Use the figure to find the values of $x$, $y$, and $z$ that make $\triangle DEF \sim \triangle HGF$.

### Explanation of the Diagram

The diagram consists of two triangles: $\triangle DEF$ and $\triangle HGF$.

- **Triangle DEF:**
- $ \angle DEF = 16^\circ $
- $ DE = 25 $
- $ DF = 24 $
- $ EF = x - 5 $

- **Triangle HGF:**
- $ \angle HGF = 2(x - 4)^\circ $
- $ GF = 14 $
- $ HF = 6z + 8 $
- $ GH = 3y $

Triangles $\triangle DEF$ and $\triangle HGF$ are given as similar ($\triangle DEF \sim \triangle HGF$), meaning corresponding angles are equal and corresponding sides are proportional.

Since the triangles are similar:
1. The corresponding angles are equal.
2. The ratios of the lengths of corresponding sides are equal.

### Steps to Find the Values of $x$, $y$, and $z$:

1. **Angle Correlations:**
- Given $\angle DEF = \angle HGF$, so:
\[ 16^\circ = 2(x - 4)^\circ \]

Solving for $x$:
\[ 16 = 2(x - 4) \]
\[ 16 = 2x - 8 \]
\[ 2x = 24 \]
\[ x = 12 \]

2. **Side Ratios:**
Since $\triangle DEF \sim \triangle HGF$:
\[ \frac{DE}{GH} = \frac{DF}{GF} = \frac{EF}{HF} \]

Using the sides:
\[ \frac{25}{3y} = \frac{24}{14} = \frac{x - 5}{6z + 8} \]

Simplifying, we get:
- For $\frac{25}{3y} = \frac{24}{14}$:
\[ 25 \cdot 14 = 24 \cd

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