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
**If \( \overline{BC} \) is tangent to Circle O, find the value of \( \overline{AB} \).**
### Diagram Description
- There is a circle labeled \( O \).
- A line segment \( \overline{BC} \) is tangent to the circle at point \( B \).
- The radius of the circle from the center \( O \) to the point of tangency \( B \) is represented as a radial line.
- There is another line segment \( \overline{AD} \) intersecting the circle at points \( A \), \( B \), and \( D \).
### Provided Values
- The length \( \overline{AB} = 8 \).
- The length \( \overline{BD} = 16 \).
### Options
a. 192
b. 13.9
c. 8
d. 24
### Explanation of Concepts
In geometry, when a line is tangent to a circle, it touches the circle at exactly one point. The radius drawn to the point of tangency is perpendicular to the tangent line.
To solve this problem, we can use the tangent-secant theorem, which states that if a tangent segment and a secant segment are drawn from a common external point, then the square of the length of the tangent segment is equal to the product of the entire length of the secant segment and the external part of the secant segment.
Using this theorem:
\[ \overline{BC}^2 = \overline{AB} \cdot \overline{BD} \]
Given that \( \overline{AB} = 8 \) and \( \overline{BD} = 16 \), plug into the equation:
\[ \overline{BC}^2 = 8 \cdot 16 = 128 \]
\[ \overline{BC} = \sqrt{128} \approx 11.3 \]
Therefore, the provided answer choices do not directly solve this exact value but ask for the missing segment \( \overline{AB} \). It would be a mistake or misinterpretation.
However, using the Radius, see AB, and given points, recheck option logic directly relative to \( r \approx 16 \), we deduce focusing actual problem AB equal.
Thus:
**Correct Answer Value: Check upon options, logically](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Febcd91cf-c92a-4a76-9c9a-e8d21c0f504d%2F1796751f-c671-4a4a-9623-ead08c2f3f3c%2Fa5e4dqd_processed.jpeg&w=3840&q=75)
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