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.
![### Finding the Tangent of Angle X
To determine the tangent of angle \( \angle X \) for the given right-angled triangle \( \triangle WXY \):
1. **Triangle Description**:
The triangle \( \triangle WXY \) is a right-angled triangle with:
- Angle \( \angle W \) being \( 90^\circ \) (right angle)
- Side \( WX \) given as \( 2\sqrt{43} \)
- Side \( WY \) given as \( \sqrt{43} \)
2. **Tangent Definition**:
In a right-angled triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In this case:
\[
\tan(\angle X) = \frac{\text{opposite}}{\text{adjacent}}
\]
For \( \angle X \):
- Opposite side to \( \angle X \) is \( WY = \sqrt{43} \)
- Adjacent side to \( \angle X \) is \( WX = 2\sqrt{43} \)
3. **Calculate the Tangent**:
\[
\tan(\angle X) = \frac{\sqrt{43}}{2\sqrt{43}} = \frac{1}{2}
\]
Thus,
\[
\tan(\angle X) = \boxed{\frac{1}{2}}
\]
The required answer is provided in simplified, rationalized form.
Do not round the answer.
### Additional Information:
- Make sure to input the exact fraction in the response box where required.
- Utilize the mathematical symbols provided (such as the square root and fraction symbols) to ensure accurate and clear notation.
By understanding this example, you'll be able to find the tangent of any angle in a right-angled triangle using the appropriate sides' lengths.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fff9a8e26-af69-42fc-821c-9ae0b96a04a7%2F094067bb-819d-42a2-ad58-53dd78675d30%2Fc1j23aq_processed.png&w=3840&q=75)
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