Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
![### Problem Statement:
Find the value of \( x \) to the nearest hundredth.
### Diagram Description:
The given image shows a right triangle with the following details:
- One leg of the right triangle, which is 10 feet long, is perpendicular to the base.
- The hypotenuse of the right triangle is 12 feet long.
- The angle opposite to the unknown length \( x \) is marked.
### Steps to Solve:
1. **Identify Known Values:**
- Opposite side: 10 ft
- Hypotenuse: 12 ft
2. **Apply the Sine Ratio:**
- In a right triangle, the sine of an angle is given by the ratio of the length of the opposite side to the length of the hypotenuse.
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
Here, \(\theta\) is the angle \( x \).
3. **Calculate Angle \( x \):**
\[
\sin(\theta) = \frac{10}{12} = \frac{5}{6} \approx 0.83
\]
Using inverse sine function (\(\sin^{-1}\)), find the angle \( x \):
\[
\theta = \sin^{-1}(0.83)
\]
This will give the angle \( x \) in degrees. Use a scientific calculator to find the value.
### Result:
After calculating the value, round \( x \) to the nearest hundredth. Type your answer in the provided box.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb5b34e65-7f01-4eff-8399-312ff1ddffb7%2F73d17920-b64f-410b-8b35-1fd93fdd90ef%2F5mhpaqw_processed.jpeg&w=3840&q=75)
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