Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section: Chapter Questions
Problem 75RE
Related questions
Question
![### Understanding the 45°-45°-90° Triangle
For an educational website on geometry, let's explore the properties of the 45°-45°-90° triangle with a quick question.
#### Question:
The hypotenuse of a 45°-45°-90° triangle measures 4 cm. What is the length of one leg of the triangle?
#### Diagram Explanation:
The provided diagram shows a right triangle where both legs are congruent, and each of the angles opposite these legs is 45°, indicating this is a 45°-45°-90° triangle. The hypotenuse, labelled as 4 cm, is opposite the right angle.
#### Answer Choices:
- A) 2 cm
- B) \( 2\sqrt{2} \) cm
- C) 4 cm
- D) \( 4\sqrt{2} \) cm
#### Solution:
In a 45°-45°-90° triangle, the legs are congruent, and each leg has a length that is \( \frac{\sqrt{2}}{2} \) times the length of the hypotenuse.
Given:
- Hypotenuse (c) = 4 cm
Calculate the length of each leg (a):
\[ a = \frac{c}{\sqrt{2}} \]
\[ a = \frac{4}{\sqrt{2}} \]
Simplify:
\[ a = 4 \cdot \frac{\sqrt{2}}{2} \]
\[ a = 2\sqrt{2} \text{ cm} \]
Thus, the correct answer is:
#### Correct Answer:
- **B) \( 2\sqrt{2} \) cm**
This example illustrates the principle that in any 45°-45°-90° triangle, the lengths of the legs are equal, and the hypotenuse is \( \sqrt{2} \) times as long as one leg.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1fefc4e4-0fc1-4d3a-92e0-1dd9cbc9acda%2F5863820b-055c-4275-8e1b-d5858d9b36a9%2F40die5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding the 45°-45°-90° Triangle
For an educational website on geometry, let's explore the properties of the 45°-45°-90° triangle with a quick question.
#### Question:
The hypotenuse of a 45°-45°-90° triangle measures 4 cm. What is the length of one leg of the triangle?
#### Diagram Explanation:
The provided diagram shows a right triangle where both legs are congruent, and each of the angles opposite these legs is 45°, indicating this is a 45°-45°-90° triangle. The hypotenuse, labelled as 4 cm, is opposite the right angle.
#### Answer Choices:
- A) 2 cm
- B) \( 2\sqrt{2} \) cm
- C) 4 cm
- D) \( 4\sqrt{2} \) cm
#### Solution:
In a 45°-45°-90° triangle, the legs are congruent, and each leg has a length that is \( \frac{\sqrt{2}}{2} \) times the length of the hypotenuse.
Given:
- Hypotenuse (c) = 4 cm
Calculate the length of each leg (a):
\[ a = \frac{c}{\sqrt{2}} \]
\[ a = \frac{4}{\sqrt{2}} \]
Simplify:
\[ a = 4 \cdot \frac{\sqrt{2}}{2} \]
\[ a = 2\sqrt{2} \text{ cm} \]
Thus, the correct answer is:
#### Correct Answer:
- **B) \( 2\sqrt{2} \) cm**
This example illustrates the principle that in any 45°-45°-90° triangle, the lengths of the legs are equal, and the hypotenuse is \( \sqrt{2} \) times as long as one leg.
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