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 3:**
In the accompanying diagram, tangent \( \overline{AB} \) and secant \( \overline{ACD} \) are drawn to circle \( O \) from point \( A \). Given that \( AB = 6 \) and \( AC = 4 \), find the length of \( AD \).
**Diagram Explanation:**
- The diagram depicts a circle with center \( O \).
- A tangent line \( \overline{AB} \) touches the circle at point \( B \).
- A secant line \( \overline{ACD} \) intersects the circle at points \( C \) and \( D \).
- The point \( A \), where both lines originate, is located outside the circle.
- The lengths \( AB \) and \( AC \) are given as 6 and 4 units, respectively. The task is to find the length of \( AD \).
**Note to students:**
To solve this problem, you might use the geometric properties of tangents and secants of a circle. Specifically, if a tangent and a secant are drawn from the same external point, then the square of the length of the tangent segment is equal to the product of the lengths of the entire secant segment and its external segment. This can be described mathematically as:
\[
AB^2 = AC \cdot AD
\]
Here, you can substitute the known values to find \( AD \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe489ebd6-5c8e-44bd-8a5f-33d2b46a9596%2Fced4bc55-852a-414f-b881-755ee99c0ef2%2F96k5gh0l_processed.jpeg&w=3840&q=75)
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