AQRS ATUV.Solve for y and z. R 30 6. 35 2.

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
icon
Related questions
Question
100%
### 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
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
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps with 2 images

Blurred answer
Knowledge Booster
Paths and Circuits
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, geometry and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
Elementary Geometry for College Students
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning