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.
![### Trigonometry Problem: Finding Angle Measures
In the right triangle ΔFGH, the measure of ∠H is 90°. The lengths of the sides are as follows: FG = 9.9 feet and HF = 14 feet. Find the measure of ∠F to the nearest tenth of a degree.
Below is the diagram of triangle ΔFGH:
```
G
|
|\
| \
9.9 | \
| \
| \
H———F
14
```
**Explanation:**
- ∠H is a right angle (90°).
- FG is the vertical side of the triangle, measuring 9.9 feet.
- HF is the horizontal side of the triangle, measuring 14 feet.
- GF is the hypotenuse of the triangle.
**Solution:**
To find the measure of ∠F, we can use the trigonometric functions. Specifically, we'll use the tangent function, which relates the opposite side to the adjacent side in a right triangle:
\[ \tan(\angle F) = \frac{\text{opposite}}{\text{adjacent}} = \frac{FG}{HF} = \frac{9.9}{14} \]
Calculate the value:
\[ \tan(\angle F) = \frac{9.9}{14} \approx 0.707 \]
Next, find the angle whose tangent is 0.707 using the arctangent (inverse tangent) function:
\[ \angle F = \tan^{-1}(0.707) \]
Using a calculator:
\[ \angle F \approx 35.2° \]
Therefore, the measure of ∠F to the nearest tenth of a degree is approximately **35.2°**.
**Answer:**
\[ \angle F = 35.2° \]
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