Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter1: The Six Trigonometric Functions
Section: Chapter Questions
Problem 2GP
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![This is a geometric problem involving a circle and a right triangle.
**Problem Statement:**
- The circle is inscribed in the right triangle.
- The triangle has side lengths labeled as follows:
- One leg of the right triangle is labeled with a length of 3.
- The other leg is labeled with a length of 4.
- The hypotenuse (longest side) is labeled with a length of 5.
- The radius of the circle is labeled as \( x \).
- The task is to find the value of \( x \).
**Diagram Description:**
- A circle is perfectly inscribed within a right triangle.
- The right triangle has its right angle at the intersection of the legs labeled 3 and 4.
- The hypotenuse, connecting the vertices of these two legs, is labeled 5.
- The circle touches all three sides of the triangle, which internally tangent to the circle.
- The radius of the circle, shown touching the right-angled vertex, is labeled \( x \).
**Additional Information for Solver:**
- The given right triangle with sides 3, 4, and 5 follows the Pythagorean theorem: \(3^2 + 4^2 = 5^2\).
- To find the radius \( x \) of an inscribed circle in a right triangle, you can use the formula for the radius \( r \) of such a circle, which is given by:
\[
r = \frac{a + b - c}{2}
\]
where \( a \) and \( b \) are the legs of the right triangle, and \( c \) is the hypotenuse.
Using this formula with the given side lengths \( a = 3 \), \( b = 4 \), and \( c = 5 \):
\[
x = \frac{3 + 4 - 5}{2} = \frac{2}{2} = 1
\]
So, the value of \( x \) is 1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F48c9b6f6-07b6-418f-ae1d-a6b2502ed423%2Fa5f57188-4082-40fc-9555-bfc5c256bc3e%2Fjf5f62f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:This is a geometric problem involving a circle and a right triangle.
**Problem Statement:**
- The circle is inscribed in the right triangle.
- The triangle has side lengths labeled as follows:
- One leg of the right triangle is labeled with a length of 3.
- The other leg is labeled with a length of 4.
- The hypotenuse (longest side) is labeled with a length of 5.
- The radius of the circle is labeled as \( x \).
- The task is to find the value of \( x \).
**Diagram Description:**
- A circle is perfectly inscribed within a right triangle.
- The right triangle has its right angle at the intersection of the legs labeled 3 and 4.
- The hypotenuse, connecting the vertices of these two legs, is labeled 5.
- The circle touches all three sides of the triangle, which internally tangent to the circle.
- The radius of the circle, shown touching the right-angled vertex, is labeled \( x \).
**Additional Information for Solver:**
- The given right triangle with sides 3, 4, and 5 follows the Pythagorean theorem: \(3^2 + 4^2 = 5^2\).
- To find the radius \( x \) of an inscribed circle in a right triangle, you can use the formula for the radius \( r \) of such a circle, which is given by:
\[
r = \frac{a + b - c}{2}
\]
where \( a \) and \( b \) are the legs of the right triangle, and \( c \) is the hypotenuse.
Using this formula with the given side lengths \( a = 3 \), \( b = 4 \), and \( c = 5 \):
\[
x = \frac{3 + 4 - 5}{2} = \frac{2}{2} = 1
\]
So, the value of \( x \) is 1.
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