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.
Solve for x.
![**4. Solve for x.**
The diagram consists of a circle intersected by a secant and a tangent. The angle formed outside the circle by the intersection of the secant and tangent is 23°. The angle formed by the intercepted arc on the circle is labeled as \((14x - 22)^\circ\).
#### Solution:
According to the properties of circles, the measure of the angle formed outside the circle by the intersection of the secant and tangent (external angle) is half the difference of the measures of the intercepted arcs.
\[ \text{Angle} = \frac{1}{2} \times (\text{Measure of larger intercepted arc} - \text{Measure of smaller intercepted arc}) \]
For our diagram:
\[ 23^\circ = \frac{1}{2} \times (14x - 22)^\circ \]
To solve for \( x \):
1. Multiply both sides by 2 to eliminate the fraction:
\[ 46^\circ = 14x - 22^\circ \]
2. Add 22° to both sides to isolate the term with \( x \):
\[ 68^\circ = 14x \]
3. Divide both sides by 14:
\[ x = \frac{68}{14} \]
\[ x = \frac{34}{7} \]
\[ x = 4.857 \]
Therefore, the value of \( x \) is approximately 4.857.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F74c34e50-fb0f-46ba-baa6-7499f059f097%2F45cae8cf-bc2c-4f43-bea9-517364d68cc3%2Fxlpo23w_processed.png&w=3840&q=75)
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