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 Overview
**Given:**
- Point \(D\) is on circle \(C\) with diameter \(AB\) as shown.
- \(AC = 6.5\) inches
- \(BD = 5\) inches
**Find:** Length of \(AD\).
### Diagram Explanation
The image displays a circle \(C\) with diameter \(AB\). Point \(D\) is located on the circle, forming a triangle \(ABD\) within the circle. \(AC\) (a segment from \(A\) to the midpoint of the diameter \(AB\)) is given as 6.5 inches and segment \(BD\) is given as 5 inches.
### Steps to Solve
1. **Determine the Lengths:**
- Since \(AC\) represents the radius of the circle, and \(C\) is the midpoint of \(AB\), the length of the radius is 6.5 inches.
- Since \(AB\) is the diameter, the length of the diameter of the circle is \(2 \times AC = 2 \times 6.5 = 13\) inches.
2. **Apply the Pythagorean Theorem:**
- Since \(D\) lies on the circle and \(ABD\) forms a right triangle with \(AB\) as its hypotenuse, we use the Pythagorean theorem.
- \(AD\) and \(BD\) form the two perpendicular sides of the right triangle \(ABD\) with hypotenuse \(AB\).
Using the Pythagorean theorem in the right triangle \(ABD\):
\[
AD^2 + BD^2 = AB^2
\]
Substituting the known values:
\[
AD^2 + 5^2 = 13^2
\]
\[
AD^2 + 25 = 169
\]
\[
AD^2 = 144
\]
\[
AD = \sqrt{144} = 12
\]
### Solution
- The length of \(AD\) is **12 inches**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5f245109-2cd9-44ee-9f70-12b58fde4a6e%2Fc76aa83a-c846-44d6-bac6-af0c91c13d39%2Fned5ey_processed.jpeg&w=3840&q=75)
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