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
![---
### Understanding Rectangles and Diagonals in Geometry
#### Problem Statement:
Consider the rectangle TREC with diagonals. We are tasked with finding the length of ET.
#### Diagram:
There is a rectangle labeled TREC with diagonals TR and EC intersecting at point X. The diagonals are given expressions as follows:
- TR is labeled as \(2x - 3\).
- EC is labeled as \(x + 7\).
#### Question:
What is the length of ET?
#### Answer Choices:
1. 10 units
2. 17 units
3. 20 units
4. 34 units
#### Explanation:
To find the length of ET, which is half of the diagonal TR (since in rectangles, diagonals bisect each other), the following steps are involved:
1. **Equate the diagonals**:
Since diagonals of a rectangle are equal:
\[
2x - 3 = x + 7
\]
2. **Solve for x**:
\[
2x - x = 7 + 3
\]
\[
x = 10
\]
3. **Substitute x back into the expressions**:
\[
TR = 2x - 3 = 2(10) - 3 = 20 - 3 = 17 \text{ units}
\]
4. **Calculate ET**:
Since diagonals of a rectangle are equal and each diagonal bisects into two equal parts:
\[
ET = \frac{TR}{2} = \frac{17}{2} = 8.5 \text{ units}
\]
However, it seems there might be a mistake in the choices given. Please check the context of study materials for correct choices. The correct solution based on steps given should be 17 units as the length of TR, not ET.
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1fefc4e4-0fc1-4d3a-92e0-1dd9cbc9acda%2F82624fee-e94a-4ca1-be47-65671e968dde%2Fp1ukl59_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
### Understanding Rectangles and Diagonals in Geometry
#### Problem Statement:
Consider the rectangle TREC with diagonals. We are tasked with finding the length of ET.
#### Diagram:
There is a rectangle labeled TREC with diagonals TR and EC intersecting at point X. The diagonals are given expressions as follows:
- TR is labeled as \(2x - 3\).
- EC is labeled as \(x + 7\).
#### Question:
What is the length of ET?
#### Answer Choices:
1. 10 units
2. 17 units
3. 20 units
4. 34 units
#### Explanation:
To find the length of ET, which is half of the diagonal TR (since in rectangles, diagonals bisect each other), the following steps are involved:
1. **Equate the diagonals**:
Since diagonals of a rectangle are equal:
\[
2x - 3 = x + 7
\]
2. **Solve for x**:
\[
2x - x = 7 + 3
\]
\[
x = 10
\]
3. **Substitute x back into the expressions**:
\[
TR = 2x - 3 = 2(10) - 3 = 20 - 3 = 17 \text{ units}
\]
4. **Calculate ET**:
Since diagonals of a rectangle are equal and each diagonal bisects into two equal parts:
\[
ET = \frac{TR}{2} = \frac{17}{2} = 8.5 \text{ units}
\]
However, it seems there might be a mistake in the choices given. Please check the context of study materials for correct choices. The correct solution based on steps given should be 17 units as the length of TR, not ET.
---
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