### Geometry Problem on Educational Concepts **Question 13 (G.CO.B.6)** --- #### Problem Statement: In the diagram below, triangle \( \triangle ABC \) with sides of 13, 15, and 16, is mapped onto \( \triangle DEF \) after a clockwise rotation of 90° about point \( P \). If \( DE = 2x - 1 \), what is the value of \( x \)? 1. \( 7 \) 2. \( 7.5 \) 3. \( 8 \) 4. \( 8.5 \) --- #### Image Description: - The original triangle \( \triangle ABC \) has side lengths labeled as follows: - \( AB = 16 \) - \( BC = 13 \) - \( AC = 15 \) - After the 90° clockwise rotation about point \( P \), the new triangle \( \triangle DEF \) is formed. - In triangle \( \triangle DEF \): - Point \( D \) is where \( A \) was - Point \( E \) is where \( B \) was - Point \( F \) is where \( C \) was - Side \( DE \) is represented as \( 2x - 1 \) --- #### Task: Given \( DE = 2x - 1 \), find the value of \( x \) by considering that \( \triangle ABC \) and \( \triangle DEF \) are congruent triangles due to the given rotation. --- #### Solution Outline: Step 1: Identify that since \( \triangle ABC \) and \( \triangle DEF \) are congruent, their corresponding sides are equal: - \( AB = DE \) - \( BC = EF \) - \( AC = DF \) Step 2: Since \( AB = 16 \), it must match \( DE \). Step 3: Set the equation: \[ 16 = 2x - 1 \] Step 4: Solve for \( x \): \[ 16 + 1 = 2x \] \[ 17 = 2x \] \[ x = \frac{17}{2} = 8.5 \] Therefore, \( x = 8.5 \). --- **Answer:** (4) 8.5
### Geometry Problem on Educational Concepts **Question 13 (G.CO.B.6)** --- #### Problem Statement: In the diagram below, triangle \( \triangle ABC \) with sides of 13, 15, and 16, is mapped onto \( \triangle DEF \) after a clockwise rotation of 90° about point \( P \). If \( DE = 2x - 1 \), what is the value of \( x \)? 1. \( 7 \) 2. \( 7.5 \) 3. \( 8 \) 4. \( 8.5 \) --- #### Image Description: - The original triangle \( \triangle ABC \) has side lengths labeled as follows: - \( AB = 16 \) - \( BC = 13 \) - \( AC = 15 \) - After the 90° clockwise rotation about point \( P \), the new triangle \( \triangle DEF \) is formed. - In triangle \( \triangle DEF \): - Point \( D \) is where \( A \) was - Point \( E \) is where \( B \) was - Point \( F \) is where \( C \) was - Side \( DE \) is represented as \( 2x - 1 \) --- #### Task: Given \( DE = 2x - 1 \), find the value of \( x \) by considering that \( \triangle ABC \) and \( \triangle DEF \) are congruent triangles due to the given rotation. --- #### Solution Outline: Step 1: Identify that since \( \triangle ABC \) and \( \triangle DEF \) are congruent, their corresponding sides are equal: - \( AB = DE \) - \( BC = EF \) - \( AC = DF \) Step 2: Since \( AB = 16 \), it must match \( DE \). Step 3: Set the equation: \[ 16 = 2x - 1 \] Step 4: Solve for \( x \): \[ 16 + 1 = 2x \] \[ 17 = 2x \] \[ x = \frac{17}{2} = 8.5 \] Therefore, \( x = 8.5 \). --- **Answer:** (4) 8.5
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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![### Geometry Problem on Educational Concepts
**Question 13 (G.CO.B.6)**
---
#### Problem Statement:
In the diagram below, triangle \( \triangle ABC \) with sides of 13, 15, and 16, is mapped onto \( \triangle DEF \) after a clockwise rotation of 90° about point \( P \).
If \( DE = 2x - 1 \), what is the value of \( x \)?
1. \( 7 \)
2. \( 7.5 \)
3. \( 8 \)
4. \( 8.5 \)
---
#### Image Description:
- The original triangle \( \triangle ABC \) has side lengths labeled as follows:
- \( AB = 16 \)
- \( BC = 13 \)
- \( AC = 15 \)
- After the 90° clockwise rotation about point \( P \), the new triangle \( \triangle DEF \) is formed.
- In triangle \( \triangle DEF \):
- Point \( D \) is where \( A \) was
- Point \( E \) is where \( B \) was
- Point \( F \) is where \( C \) was
- Side \( DE \) is represented as \( 2x - 1 \)
---
#### Task:
Given \( DE = 2x - 1 \), find the value of \( x \) by considering that \( \triangle ABC \) and \( \triangle DEF \) are congruent triangles due to the given rotation.
---
#### Solution Outline:
Step 1:
Identify that since \( \triangle ABC \) and \( \triangle DEF \) are congruent, their corresponding sides are equal:
- \( AB = DE \)
- \( BC = EF \)
- \( AC = DF \)
Step 2:
Since \( AB = 16 \), it must match \( DE \).
Step 3:
Set the equation:
\[ 16 = 2x - 1 \]
Step 4:
Solve for \( x \):
\[ 16 + 1 = 2x \]
\[ 17 = 2x \]
\[ x = \frac{17}{2} = 8.5 \]
Therefore, \( x = 8.5 \).
---
**Answer:** (4) 8.5](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5a150af8-9e8d-4015-9821-abf8efa8b3eb%2F667100bb-9da8-4bb1-a9fc-785f89103885%2Fwbjrkk.jpeg&w=3840&q=75)
Transcribed Image Text:### Geometry Problem on Educational Concepts
**Question 13 (G.CO.B.6)**
---
#### Problem Statement:
In the diagram below, triangle \( \triangle ABC \) with sides of 13, 15, and 16, is mapped onto \( \triangle DEF \) after a clockwise rotation of 90° about point \( P \).
If \( DE = 2x - 1 \), what is the value of \( x \)?
1. \( 7 \)
2. \( 7.5 \)
3. \( 8 \)
4. \( 8.5 \)
---
#### Image Description:
- The original triangle \( \triangle ABC \) has side lengths labeled as follows:
- \( AB = 16 \)
- \( BC = 13 \)
- \( AC = 15 \)
- After the 90° clockwise rotation about point \( P \), the new triangle \( \triangle DEF \) is formed.
- In triangle \( \triangle DEF \):
- Point \( D \) is where \( A \) was
- Point \( E \) is where \( B \) was
- Point \( F \) is where \( C \) was
- Side \( DE \) is represented as \( 2x - 1 \)
---
#### Task:
Given \( DE = 2x - 1 \), find the value of \( x \) by considering that \( \triangle ABC \) and \( \triangle DEF \) are congruent triangles due to the given rotation.
---
#### Solution Outline:
Step 1:
Identify that since \( \triangle ABC \) and \( \triangle DEF \) are congruent, their corresponding sides are equal:
- \( AB = DE \)
- \( BC = EF \)
- \( AC = DF \)
Step 2:
Since \( AB = 16 \), it must match \( DE \).
Step 3:
Set the equation:
\[ 16 = 2x - 1 \]
Step 4:
Solve for \( x \):
\[ 16 + 1 = 2x \]
\[ 17 = 2x \]
\[ x = \frac{17}{2} = 8.5 \]
Therefore, \( x = 8.5 \).
---
**Answer:** (4) 8.5
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