Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
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ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![### Determine the Length of Sides \( x \) and \( y \) in Each Triangle
In the image presented, there is a right-angled triangle with the following angles and side lengths:
- One angle measures \( 30^\circ \).
- Another angle measures \( 60^\circ \).
- The hypotenuse is labeled as \( 28 \) units.
- The side opposite the \( 30^\circ \) angle is labeled as \( y \).
- The side opposite the \( 60^\circ \) angle is labeled as \( x \).
### Detailed Explanation
In a 30-60-90 triangle (a special right triangle), the side lengths follow a specific ratio which can be used to determine the unknown sides. The relationships between the sides can be summarized as:
- The length of the side opposite the \( 30^\circ \) angle (shortest side) is \( \frac{1}{2} \) of the hypotenuse.
- The length of the side opposite the \( 60^\circ \) angle is \( \frac{\sqrt{3}}{2} \) of the hypotenuse.
Given that the hypotenuse is 28 units, we can find the lengths of \( x \) and \( y \) by applying these relationships:
1. **Finding \( y \) (opposite the \( 30^\circ \) angle):**
\[
y = \frac{1}{2} \times \text{hypotenuse} = \frac{1}{2} \times 28 = 14 \text{ units}
\]
2. **Finding \( x \) (opposite the \( 60^\circ \) angle):**
\[
x = \frac{\sqrt{3}}{2} \times \text{hypotenuse} = \frac{\sqrt{3}}{2} \times 28 = 14\sqrt{3} \text{ units}
\]
Therefore, the side lengths are:
- \( y = 14 \) units
- \( x = 14\sqrt{3} \) units](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc4e58655-e53f-42ee-b39f-787912eba8a6%2F6a3e865b-180b-4814-91ab-13221dc93d6d%2Fwx3pw3i_processed.png&w=3840&q=75)
Transcribed Image Text:### Determine the Length of Sides \( x \) and \( y \) in Each Triangle
In the image presented, there is a right-angled triangle with the following angles and side lengths:
- One angle measures \( 30^\circ \).
- Another angle measures \( 60^\circ \).
- The hypotenuse is labeled as \( 28 \) units.
- The side opposite the \( 30^\circ \) angle is labeled as \( y \).
- The side opposite the \( 60^\circ \) angle is labeled as \( x \).
### Detailed Explanation
In a 30-60-90 triangle (a special right triangle), the side lengths follow a specific ratio which can be used to determine the unknown sides. The relationships between the sides can be summarized as:
- The length of the side opposite the \( 30^\circ \) angle (shortest side) is \( \frac{1}{2} \) of the hypotenuse.
- The length of the side opposite the \( 60^\circ \) angle is \( \frac{\sqrt{3}}{2} \) of the hypotenuse.
Given that the hypotenuse is 28 units, we can find the lengths of \( x \) and \( y \) by applying these relationships:
1. **Finding \( y \) (opposite the \( 30^\circ \) angle):**
\[
y = \frac{1}{2} \times \text{hypotenuse} = \frac{1}{2} \times 28 = 14 \text{ units}
\]
2. **Finding \( x \) (opposite the \( 60^\circ \) angle):**
\[
x = \frac{\sqrt{3}}{2} \times \text{hypotenuse} = \frac{\sqrt{3}}{2} \times 28 = 14\sqrt{3} \text{ units}
\]
Therefore, the side lengths are:
- \( y = 14 \) units
- \( x = 14\sqrt{3} \) units
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