### Lesson: Understanding Reflections and Proportionality#### Problem Statement:Malik, who is 5 feet tall, stands 15 feet from a mirror. In the mirror, he can see the reflection of the top of a tree which he measures to be 92 feet away from the mirror.#### Diagram Description:The diagram shows a triangular representation with Malik, the tree, and the mirror. Malik is indicated at the base of the triangle standing 15 feet away from the mirror, and the tree's top is visible in the mirror at a measured distance of 92 feet.#### Questions:**Part A:** Determine the height of the tree pictured in the diagram above. Round your answer to the nearest foot if applicable. Use mathematics to explain how you determined your answer.**Part B:** How much taller is the tree than Malik? Explain your reasoning.### Detailed Explanation:In the given diagram, the setup creates similar triangles due to the reflection property.#### Step-by-Step Solution:**Part A:**1. Identify the proportions using the properties of similar triangles.2. Since the triangles are similar, the ratio of corresponding sides is the same.3. Set up the proportion: - Malik's distance to the mirror: Height of Malik = Distance to tree reflection: Height of Tree - (15 feet)/(5 feet) = (92 feet)/(Height of Tree)4. Solve for the height of the tree: \[ \frac{15}{5} = \frac{92}{\text{Height of Tree}} \implies 3 = \frac{92}{\text{Height of Tree}} \] Thus, Height of Tree ≈ $\frac{92}{3} $ ≈ 30.67 ≈ 31 feet.**Part B:**1. Calculate the difference in height between Malik and the tree: - Tree height: 31 feet - Malik's height: 5 feet - Difference in height: 31 feet - 5 feet = 26 feet2. Therefore, the tree is 26 feet taller than Malik.**Summary:**Using similar triangles and proportions, we determined that the height of the tree is approximately 31 feet. Compared to Malik's height of 5 feet, the tree is 26 feet taller.---This practical example aids in understanding the application of similar triangles and proportionality in geometry.

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Malik, who is 5 feet tall, stands 15 feet from a mirror In the mirror, he can see the refliection of the top of a tree, which he measures to be 92 feet away from the miror. Part A Deleimine the height of the tree pietured in the diagram above Round your answer to the nearest foot if applicable Use mathematics to explain how you determine your answer Munor Part B How much taller is the tree than Malık? Explain your Feasonng

Malik, who is 5 feet tall, stands 15 feet from a mirror In the mirror, he can see the refliection of the top of a tree, which he measures to be 92 feet away from the miror. Part A Deleimine the height of the tree pietured in the diagram above Round your answer to the nearest foot if applicable Use mathematics to explain how you determine your answer Munor Part B How much taller is the tree than Malık? Explain your Feasonng

Elementary Algebra

17th Edition

ISBN:9780998625713

Author:Lynn Marecek, MaryAnne Anthony-Smith

Publisher:Lynn Marecek, MaryAnne Anthony-Smith

Chapter10: Quadratic Equations

Section10.4: Solve Applications Modeled By Quadratic Equations

Problem 160E: A city planner wants to build a bridge across a lake in a park. To find the length of the bridge, he...

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Question

### Lesson: Understanding Reflections and Proportionality

#### Problem Statement:
Malik, who is 5 feet tall, stands 15 feet from a mirror. In the mirror, he can see the reflection of the top of a tree which he measures to be 92 feet away from the mirror.

#### Diagram Description:
The diagram shows a triangular representation with Malik, the tree, and the mirror. Malik is indicated at the base of the triangle standing 15 feet away from the mirror, and the tree's top is visible in the mirror at a measured distance of 92 feet.

#### Questions:
**Part A:** Determine the height of the tree pictured in the diagram above. Round your answer to the nearest foot if applicable. Use mathematics to explain how you determined your answer.

**Part B:** How much taller is the tree than Malik? Explain your reasoning.

### Detailed Explanation:
In the given diagram, the setup creates similar triangles due to the reflection property.

#### Step-by-Step Solution:

**Part A:**
1. Identify the proportions using the properties of similar triangles.
2. Since the triangles are similar, the ratio of corresponding sides is the same.
3. Set up the proportion:
- Malik's distance to the mirror: Height of Malik = Distance to tree reflection: Height of Tree
- (15 feet)/(5 feet) = (92 feet)/(Height of Tree)
4. Solve for the height of the tree:
\[
\frac{15}{5} = \frac{92}{\text{Height of Tree}} \implies 3 = \frac{92}{\text{Height of Tree}}
\]
Thus, Height of Tree ≈ $\frac{92}{3} $ ≈ 30.67 ≈ 31 feet.

**Part B:**
1. Calculate the difference in height between Malik and the tree:
- Tree height: 31 feet
- Malik's height: 5 feet
- Difference in height: 31 feet - 5 feet = 26 feet
2. Therefore, the tree is 26 feet taller than Malik.

**Summary:**
Using similar triangles and proportions, we determined that the height of the tree is approximately 31 feet. Compared to Malik's height of 5 feet, the tree is 26 feet taller.

---
This practical example aids in understanding the application of similar triangles and proportionality in geometry.

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