B K5 7- 12 Note: Figure not drawn to scale. The area of triangle ABC above is at least 48 but no more than 60. If y is an integer, what is one possible value of x ?
B K5 7- 12 Note: Figure not drawn to scale. The area of triangle ABC above is at least 48 but no more than 60. If y is an integer, what is one possible value of x ?
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
Related questions
Question
![### Triangle ABC Problem
**Diagram Explanation:**
The diagram depicts a right triangle \( \triangle ABC \) with:
- \( AB = y \) (the vertical side)
- \( BC = x \) (the horizontal side forming a right angle with \( AC \))
- \( AC = 12 \), divided into two segments \( AD = 5 \) and \( DC = 7 \)
**Problem Statement:**
> The area of triangle \( ABC \) is at least 48 but no more than 60. If \( y \) is an integer, what is one possible value of \( x \)?
**Solution Approach:**
The area of a right triangle is calculated by the formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here:
- The base \( AC = 12 \)
- The height \( y \)
Using the area constraints \( 48 \leq \text{Area} \leq 60 \), solve for \( y \):
\[
48 \leq \frac{1}{2} \times 12 \times y \leq 60
\]
\[
48 \leq 6y \leq 60
\]
Divide the entire inequality by 6:
\[
8 \leq y \leq 10
\]
Since \( y \) is an integer, possible values for \( y \) are 8, 9, or 10.
Next, since \( y^2 = x^2 + 12^2 \), solve for \( x \) using one of these potential \( y \) values.
Example for \( y = 9 \):
\[
9^2 = x^2 + 12^2
\]
\[
81 = x^2 + 144
\]
\[
x^2 = 81 - 144
\]
\[
x^2 = 225
\]
\[
x = \sqrt{225} = 15
\]
Thus, one possible value of \( x \) is 15.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4b5ba176-efae-4d30-9d89-522676c515dc%2Fcf23786b-4c4f-4791-a16f-ce7edbb742d2%2Fye50yfj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Triangle ABC Problem
**Diagram Explanation:**
The diagram depicts a right triangle \( \triangle ABC \) with:
- \( AB = y \) (the vertical side)
- \( BC = x \) (the horizontal side forming a right angle with \( AC \))
- \( AC = 12 \), divided into two segments \( AD = 5 \) and \( DC = 7 \)
**Problem Statement:**
> The area of triangle \( ABC \) is at least 48 but no more than 60. If \( y \) is an integer, what is one possible value of \( x \)?
**Solution Approach:**
The area of a right triangle is calculated by the formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here:
- The base \( AC = 12 \)
- The height \( y \)
Using the area constraints \( 48 \leq \text{Area} \leq 60 \), solve for \( y \):
\[
48 \leq \frac{1}{2} \times 12 \times y \leq 60
\]
\[
48 \leq 6y \leq 60
\]
Divide the entire inequality by 6:
\[
8 \leq y \leq 10
\]
Since \( y \) is an integer, possible values for \( y \) are 8, 9, or 10.
Next, since \( y^2 = x^2 + 12^2 \), solve for \( x \) using one of these potential \( y \) values.
Example for \( y = 9 \):
\[
9^2 = x^2 + 12^2
\]
\[
81 = x^2 + 144
\]
\[
x^2 = 81 - 144
\]
\[
x^2 = 225
\]
\[
x = \sqrt{225} = 15
\]
Thus, one possible value of \( x \) is 15.
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