### Problem Statement In the given diagram, two triangles are shown: ΔABC and ΔCDE. Here, AB is 3 units, BC is 4.5 units, DE is \( x \) units, and CD is \( x + 6 \) units. The lengths marked for these line segments are given as: - \( AB = 3 \) - \( BC = 4.5 \) - \( DE = x \) - \( CD = x + 6 \) The triangles share a common vertex at C, and lines EA and EB are red marked indicating they are congruent. Your goal is to **find the length of CD**. ### Steps to Solve 1. **Recognize Similar Triangles**: From the diagram assumptions, ΔABC is similar to ΔCDE by AA similarity (both triangles share ∠EAD and ∠ACD). 2. **Set Up Proportions**: Since the triangles are similar, the ratios of corresponding sides in similar triangles are equal. \[ \frac{AB}{DE} = \frac{BC}{CD} \] Substitute the given lengths: \[ \frac{3}{x} = \frac{4.5}{x + 6} \] 3. **Cross-Multiply to Solve for \( x \)**: \[ 3(x + 6) = 4.5x \] Simplify the equation. \[ 3x + 18 = 4.5x \] \[ 18 = 4.5x - 3x \] \[ 18 = 1.5x \] \[ x = 12 \] 4. **Find CD**: Substitute the value of \( x \) to find the length of CD. \[ CD = x + 6 = 12 + 6 = 18 \] ### Conclusion The length of CD is \( \boxed{18} \) units.
### Problem Statement In the given diagram, two triangles are shown: ΔABC and ΔCDE. Here, AB is 3 units, BC is 4.5 units, DE is \( x \) units, and CD is \( x + 6 \) units. The lengths marked for these line segments are given as: - \( AB = 3 \) - \( BC = 4.5 \) - \( DE = x \) - \( CD = x + 6 \) The triangles share a common vertex at C, and lines EA and EB are red marked indicating they are congruent. Your goal is to **find the length of CD**. ### Steps to Solve 1. **Recognize Similar Triangles**: From the diagram assumptions, ΔABC is similar to ΔCDE by AA similarity (both triangles share ∠EAD and ∠ACD). 2. **Set Up Proportions**: Since the triangles are similar, the ratios of corresponding sides in similar triangles are equal. \[ \frac{AB}{DE} = \frac{BC}{CD} \] Substitute the given lengths: \[ \frac{3}{x} = \frac{4.5}{x + 6} \] 3. **Cross-Multiply to Solve for \( x \)**: \[ 3(x + 6) = 4.5x \] Simplify the equation. \[ 3x + 18 = 4.5x \] \[ 18 = 4.5x - 3x \] \[ 18 = 1.5x \] \[ x = 12 \] 4. **Find CD**: Substitute the value of \( x \) to find the length of CD. \[ CD = x + 6 = 12 + 6 = 18 \] ### Conclusion The length of CD is \( \boxed{18} \) units.
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
Related questions
Question
![### Problem Statement
In the given diagram, two triangles are shown: ΔABC and ΔCDE. Here, AB is 3 units, BC is 4.5 units, DE is \( x \) units, and CD is \( x + 6 \) units. The lengths marked for these line segments are given as:
- \( AB = 3 \)
- \( BC = 4.5 \)
- \( DE = x \)
- \( CD = x + 6 \)
The triangles share a common vertex at C, and lines EA and EB are red marked indicating they are congruent.
Your goal is to **find the length of CD**.
### Steps to Solve
1. **Recognize Similar Triangles**:
From the diagram assumptions, ΔABC is similar to ΔCDE by AA similarity (both triangles share ∠EAD and ∠ACD).
2. **Set Up Proportions**:
Since the triangles are similar, the ratios of corresponding sides in similar triangles are equal.
\[
\frac{AB}{DE} = \frac{BC}{CD}
\]
Substitute the given lengths:
\[
\frac{3}{x} = \frac{4.5}{x + 6}
\]
3. **Cross-Multiply to Solve for \( x \)**:
\[
3(x + 6) = 4.5x
\]
Simplify the equation.
\[
3x + 18 = 4.5x
\]
\[
18 = 4.5x - 3x
\]
\[
18 = 1.5x
\]
\[
x = 12
\]
4. **Find CD**:
Substitute the value of \( x \) to find the length of CD.
\[
CD = x + 6 = 12 + 6 = 18
\]
### Conclusion
The length of CD is \( \boxed{18} \) units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fde328eaf-c76e-4ecf-b505-50cb9ab0f03b%2Fe9e16c1c-384b-4577-9a37-8cc6d4ce48cd%2Fv772929.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
In the given diagram, two triangles are shown: ΔABC and ΔCDE. Here, AB is 3 units, BC is 4.5 units, DE is \( x \) units, and CD is \( x + 6 \) units. The lengths marked for these line segments are given as:
- \( AB = 3 \)
- \( BC = 4.5 \)
- \( DE = x \)
- \( CD = x + 6 \)
The triangles share a common vertex at C, and lines EA and EB are red marked indicating they are congruent.
Your goal is to **find the length of CD**.
### Steps to Solve
1. **Recognize Similar Triangles**:
From the diagram assumptions, ΔABC is similar to ΔCDE by AA similarity (both triangles share ∠EAD and ∠ACD).
2. **Set Up Proportions**:
Since the triangles are similar, the ratios of corresponding sides in similar triangles are equal.
\[
\frac{AB}{DE} = \frac{BC}{CD}
\]
Substitute the given lengths:
\[
\frac{3}{x} = \frac{4.5}{x + 6}
\]
3. **Cross-Multiply to Solve for \( x \)**:
\[
3(x + 6) = 4.5x
\]
Simplify the equation.
\[
3x + 18 = 4.5x
\]
\[
18 = 4.5x - 3x
\]
\[
18 = 1.5x
\]
\[
x = 12
\]
4. **Find CD**:
Substitute the value of \( x \) to find the length of CD.
\[
CD = x + 6 = 12 + 6 = 18
\]
### Conclusion
The length of CD is \( \boxed{18} \) units.
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