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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![### Solving for \( y \) and \( z \) in Similar Triangles
Given:
\[
\triangle QRS \sim \triangle TUV
\]
We are asked to solve for \( y \) and \( z \).
#### Diagram Description:
- **Triangle \( QRS \)**:
- Right Triangle
- Side \( QS \) = 2 units
- Side \( QR \) = 6 units
- Hypotenuse \( SR \) = \( z \) units
- **Triangle \( TUV \)**:
- Right Triangle
- Side \( UT \) = 30 units
- Side \( TV \) = \( y \) units
- Hypotenuse \( UV \) = 35 units
The notation \( \triangle QRS \sim \triangle TUV \) indicates that the triangles are similar, meaning their corresponding angles are equal and their corresponding sides are proportional.
#### Step-by-Step Solution:
1. **Identify Corresponding Sides:**
Since the triangles are similar, the ratios of the corresponding sides are equal:
\[
\frac{QS}{UT} = \frac{QR}{TV} = \frac{SR}{UV}
\]
2. **Using Side \( QS \) and \( UT \):**
\[
\frac{QS}{UT} = \frac{2}{30} = \frac{1}{15}
\]
3. **Using Side \( QR \) and \( TV \):**
\[
\frac{QR}{TV} = \frac{6}{y}
\]
Since the ratio \(\frac{1}{15}\) must be equal to \(\frac{6}{y}\):
\[
\frac{1}{15} = \frac{6}{y}
\]
Solving for \( y \):
\[
y \cdot 1 = 15 \cdot 6 \implies y = 90
\]
4. **Using Hypotenuse \( SR \) and \( UV \):**
\[
\frac{SR}{UV} = \frac{z}{35}
\]
Since the ratio \(\frac{1}{15}\) must be equal to \(\frac{z}{35}\):
\[
\frac{1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff5cb9589-b2ce-4a97-94e0-f1b9f87178b8%2F0d60c7ec-5c67-460a-acd3-d910cb5a780c%2Fx1p6gr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving for \( y \) and \( z \) in Similar Triangles
Given:
\[
\triangle QRS \sim \triangle TUV
\]
We are asked to solve for \( y \) and \( z \).
#### Diagram Description:
- **Triangle \( QRS \)**:
- Right Triangle
- Side \( QS \) = 2 units
- Side \( QR \) = 6 units
- Hypotenuse \( SR \) = \( z \) units
- **Triangle \( TUV \)**:
- Right Triangle
- Side \( UT \) = 30 units
- Side \( TV \) = \( y \) units
- Hypotenuse \( UV \) = 35 units
The notation \( \triangle QRS \sim \triangle TUV \) indicates that the triangles are similar, meaning their corresponding angles are equal and their corresponding sides are proportional.
#### Step-by-Step Solution:
1. **Identify Corresponding Sides:**
Since the triangles are similar, the ratios of the corresponding sides are equal:
\[
\frac{QS}{UT} = \frac{QR}{TV} = \frac{SR}{UV}
\]
2. **Using Side \( QS \) and \( UT \):**
\[
\frac{QS}{UT} = \frac{2}{30} = \frac{1}{15}
\]
3. **Using Side \( QR \) and \( TV \):**
\[
\frac{QR}{TV} = \frac{6}{y}
\]
Since the ratio \(\frac{1}{15}\) must be equal to \(\frac{6}{y}\):
\[
\frac{1}{15} = \frac{6}{y}
\]
Solving for \( y \):
\[
y \cdot 1 = 15 \cdot 6 \implies y = 90
\]
4. **Using Hypotenuse \( SR \) and \( UV \):**
\[
\frac{SR}{UV} = \frac{z}{35}
\]
Since the ratio \(\frac{1}{15}\) must be equal to \(\frac{z}{35}\):
\[
\frac{1
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