Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
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:** What are the exact side lengths of the triangle shown?
**Description of the Diagram:**
The diagram illustrates a right-angled triangle with one of the angles being 45 degrees, meaning it is a 45°-45°-90° triangle. The sides of the triangle are labeled as follows:
- The hypotenuse is labeled as "c."
- One leg of the triangle is labeled "a."
- The other leg is given a specific length of 8 units.
In a 45°-45°-90° triangle, the legs are congruent, and the hypotenuse is \(a \sqrt{2}\) times the length of a leg. Given one leg is 8 units (both legs are thus equal), the calculations proceed accordingly.
**Solution:**
1. Both legs of the triangle (since it is isosceles): \( a = 8 \) units.
2. Using the relationship \( c = a \sqrt{2} \), where \( a = 8 \):
\[ c = 8 \sqrt{2} \]
Therefore, the exact side lengths of the triangle are:
- Leg \( a \) = 8 units
- Leg \( b \) = 8 units (since \( a = b \))
- Hypotenuse \( c = 8 \sqrt{2} \) units](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc4e58655-e53f-42ee-b39f-787912eba8a6%2F58c1dd8f-af9e-4c25-9d89-0d6f8d4aa7c4%2Fdngcpk_processed.png&w=3840&q=75)
Transcribed Image Text:**Question:** What are the exact side lengths of the triangle shown?
**Description of the Diagram:**
The diagram illustrates a right-angled triangle with one of the angles being 45 degrees, meaning it is a 45°-45°-90° triangle. The sides of the triangle are labeled as follows:
- The hypotenuse is labeled as "c."
- One leg of the triangle is labeled "a."
- The other leg is given a specific length of 8 units.
In a 45°-45°-90° triangle, the legs are congruent, and the hypotenuse is \(a \sqrt{2}\) times the length of a leg. Given one leg is 8 units (both legs are thus equal), the calculations proceed accordingly.
**Solution:**
1. Both legs of the triangle (since it is isosceles): \( a = 8 \) units.
2. Using the relationship \( c = a \sqrt{2} \), where \( a = 8 \):
\[ c = 8 \sqrt{2} \]
Therefore, the exact side lengths of the triangle are:
- Leg \( a \) = 8 units
- Leg \( b \) = 8 units (since \( a = b \))
- Hypotenuse \( c = 8 \sqrt{2} \) units
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