Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Understanding Similar Triangles: Problem Solving
#### Problem Statement:
Consider the following figure where \( \triangle ABC \sim \triangle DBE \).
#### Diagram Explanation:
- The diagram shows two triangles, \( \triangle ABC \) and \( \triangle DBE \), where the smaller triangle \( \triangle DBE \) is similar to the larger triangle \( \triangle ABC \).
- The points in the diagram are labeled as follows:
- \( A \) is one vertex of the larger triangle.
- \( B \) is the top vertex common to both triangles.
- \( C \) is the far-right vertex of the larger triangle.
- \( D \) is a point on \( AC \).
- \( E \) is a point on \( BC \) such that \( E \) is the midpoint of \( CB \).
#### Given Information:
- Length of \( AC = 16 \)
- Length of \( CB = 10 \)
- Point \( E \) is the midpoint of \( CB \).
#### Find:
- The length of segment \( DE \).
#### Hint:
- Let \( DE = x \), and solve an equation.
#### Solution Process:
1. Since \( \triangle ABC \sim \triangle DBE \), the triangles are similar by AA criterion (Angle-Angle). This means their corresponding sides are proportional.
2. Midpoint means that each segment from the midpoint divides the side into two equal lengths.
3. Length of \( CE \) is \( \frac{CB}{2} = \frac{10}{2} = 5 \).
4. Determine the proportionality constant (scaling factor) by looking at \( \frac{DE}{AC} \):
- Since \( AC \) of \( \triangle ABC \) becomes \( DE \) of \( \triangle DBE \), and \( \triangle DBE \) is half the size of \( \triangle ABC \) on each side, because it’s determined through midpoint.
- Therefore, \( DE = \frac{AC}{2} = \frac{16}{2} = 8 \).
Thus, the length of \( DE \) is:
\[ DE = 8 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf8dd5bb-1368-4a10-9929-9294802d0b74%2Fa9f856cf-1c8b-426f-b718-6a4becc3cc84%2F7jkjdq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Similar Triangles: Problem Solving
#### Problem Statement:
Consider the following figure where \( \triangle ABC \sim \triangle DBE \).
#### Diagram Explanation:
- The diagram shows two triangles, \( \triangle ABC \) and \( \triangle DBE \), where the smaller triangle \( \triangle DBE \) is similar to the larger triangle \( \triangle ABC \).
- The points in the diagram are labeled as follows:
- \( A \) is one vertex of the larger triangle.
- \( B \) is the top vertex common to both triangles.
- \( C \) is the far-right vertex of the larger triangle.
- \( D \) is a point on \( AC \).
- \( E \) is a point on \( BC \) such that \( E \) is the midpoint of \( CB \).
#### Given Information:
- Length of \( AC = 16 \)
- Length of \( CB = 10 \)
- Point \( E \) is the midpoint of \( CB \).
#### Find:
- The length of segment \( DE \).
#### Hint:
- Let \( DE = x \), and solve an equation.
#### Solution Process:
1. Since \( \triangle ABC \sim \triangle DBE \), the triangles are similar by AA criterion (Angle-Angle). This means their corresponding sides are proportional.
2. Midpoint means that each segment from the midpoint divides the side into two equal lengths.
3. Length of \( CE \) is \( \frac{CB}{2} = \frac{10}{2} = 5 \).
4. Determine the proportionality constant (scaling factor) by looking at \( \frac{DE}{AC} \):
- Since \( AC \) of \( \triangle ABC \) becomes \( DE \) of \( \triangle DBE \), and \( \triangle DBE \) is half the size of \( \triangle ABC \) on each side, because it’s determined through midpoint.
- Therefore, \( DE = \frac{AC}{2} = \frac{16}{2} = 8 \).
Thus, the length of \( DE \) is:
\[ DE = 8 \]
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