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.
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### Geometry Unit 4: End of Unit Summary
#### Problem 5:
**Objective:** Find the value of \(x\). Round to the nearest tenth.
A right-angled triangle is shown with an acute angle of \(37^\circ\). The side opposite this angle is labeled \(x\), and the adjacent side is labeled \(17.3\).
**Triangle Diagram:**
```
/|
/ |
x / | 17.3
/ 37°|
/_____|
```
**Task:**
- Find the value of \(x\).
- Show your work in the provided field.
**Solution:**
- To find the value of \(x\), we will use trigonometric ratios. Specifically, we'll use the tangent function, which relates the opposite side and the adjacent side of a right triangle.
\[ \tan(37^\circ) = \frac{\text{opposite}}{\text{adjacent}} \]
Given:
- Opposite side = \(x\)
- Adjacent side = \(17.3\)
The equation becomes:
\[ \tan(37^\circ) = \frac{x}{17.3} \]
Solving for \(x\):
\[ x = 17.3 \times \tan(37^\circ) \]
Use a calculator to find \(\tan(37^\circ)\):
\[ \tan(37^\circ) \approx 0.7536 \]
Therefore:
\[ x = 17.3 \times 0.7536 \approx 13.0 \]
So, \(x \approx 13.0\) (rounded to the nearest tenth).
**Show Your Work:**
\[ x = 17.3 \times \tan(37^\circ) \]
\[ x \approx 17.3 \times 0.7536 \]
\[ x \approx 13.0 \]
---
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- **0 pts:** Incomplete
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- **2 pts:** Between tier 2 and tier 3 Response
- **2.5 pts:** Tier 2 Response
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