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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

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

Elementary Geometry For College Students, 7e
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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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
Transcribed Image Text:### 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
Transcribed Image Text:### 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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