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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Question
![### Geometric Problem: Length of a Triangle Side
#### Problem Statement:
The diagram above is not drawn accurately. Find the length of \( x \).
#### Diagram Description:
The given diagram consists of a right-angled triangle with an internal right-angled triangle. The dimensions provided are as follows:
- The height of the larger triangle is \( 12 \) cm.
- The height of the smaller, internal triangle is \( 4 \) cm.
- The base of the larger triangle is divided into two segments: \( 8 \) cm and \( x \) cm.
### Explanation:
To find the length of \( x \), we can use the property of similar triangles. The triangles are similar if the angles are equal. Since both triangles share a right angle and a common angle, they are similar by AA similarity criterion.
Let's denote:
- The height of the smaller triangle as \( h_1 = 4 \) cm.
- The height of the larger triangle as \( h_2 = 12 \) cm.
- The base of the smaller triangle as \( b_1 = 8 \) cm.
- The base of the larger triangle as \( b_1 + x \).
Since the triangles are similar:
\[
\frac{4}{12} = \frac{8}{8 + x}
\]
Simplify the left side:
\[
\frac{1}{3} = \frac{8}{8 + x}
\]
Cross-multiplying gives:
\[
1 \cdot (8 + x) = 3 \cdot 8
\]
Which simplifies to:
\[
8 + x = 24
\]
\[
x = 24 - 8
\]
\[
x = 16
\]
So, the length of \( x \) is 16 cm.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa0f40088-de79-422a-8e62-fd0321faa220%2F6725ff2b-0286-4c44-ace3-d81254a2cea4%2Fus5e0fk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Geometric Problem: Length of a Triangle Side
#### Problem Statement:
The diagram above is not drawn accurately. Find the length of \( x \).
#### Diagram Description:
The given diagram consists of a right-angled triangle with an internal right-angled triangle. The dimensions provided are as follows:
- The height of the larger triangle is \( 12 \) cm.
- The height of the smaller, internal triangle is \( 4 \) cm.
- The base of the larger triangle is divided into two segments: \( 8 \) cm and \( x \) cm.
### Explanation:
To find the length of \( x \), we can use the property of similar triangles. The triangles are similar if the angles are equal. Since both triangles share a right angle and a common angle, they are similar by AA similarity criterion.
Let's denote:
- The height of the smaller triangle as \( h_1 = 4 \) cm.
- The height of the larger triangle as \( h_2 = 12 \) cm.
- The base of the smaller triangle as \( b_1 = 8 \) cm.
- The base of the larger triangle as \( b_1 + x \).
Since the triangles are similar:
\[
\frac{4}{12} = \frac{8}{8 + x}
\]
Simplify the left side:
\[
\frac{1}{3} = \frac{8}{8 + x}
\]
Cross-multiplying gives:
\[
1 \cdot (8 + x) = 3 \cdot 8
\]
Which simplifies to:
\[
8 + x = 24
\]
\[
x = 24 - 8
\]
\[
x = 16
\]
So, the length of \( x \) is 16 cm.
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