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.
![## Instruction
Solve for \( x \). Show all work.
## Diagram Explanation
In the diagram, we have a circle with two intersecting chords. The lengths of the segments of the chords are given as follows:
- One chord is divided into segments of lengths 15 and 9.
- The other chord is divided into segments of lengths 6 and \( x \).
## Solution Description
To solve for \( x \) in this context, we will use the intersecting chords theorem, which states that for two intersecting chords in a circle, the products of the lengths of the segments of each chord are equal.
Mathematically, this is expressed as:
\[ (part_1 \times part_2) = (part_3 \times part_4) \]
Given the lengths from the diagram:
\[ 15 \times 9 = 6 \times x \]
First, calculate the product on the left side:
\[ 15 \times 9 = 135 \]
Set up the equation:
\[ 135 = 6 \times x \]
Now, solve for \( x \):
\[ x = \frac{135}{6} \]
Simplify the fraction:
\[ x = 22.5 \]
Therefore, the value of \( x \) is 22.5.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F32100c6a-eba5-49dd-834d-e3d8e02c9e26%2F6975f19f-102b-4aa7-a109-38b14eea0070%2Fr3r40jp_processed.png&w=3840&q=75)
![### Geometry Problem
**Problem Statement:**
Given a circle with points \( A \), \( B \), and \( C \) on its circumference, where point \( A \) is also the point of tangency with the tangent line, and \( \angle CAB \) is given as \( 101^\circ \). You are required to find the measure of the arc \( AB \).
**Diagram Explanation:**
- The circle shown has points \( A \) and \( B \) on its circumference.
- Point \( C \) is another point on the circumference such that line segment \( CA \) is drawn.
- \( \angle CAB \) is an inscribed angle which measures \( 101^\circ \).
- There is a tangent line at point \( A \).
**Multiple Choice Options:**
1. \( 202^\circ \)
2. \( 180^\circ \)
3. \( 50.5^\circ \)
4. \( 101^\circ \)
**Solution:**
In a circle, the measure of the arc intercepted by an inscribed angle is twice the measure of the angle. Therefore, if \( \angle CAB \) measures \( 101^\circ \), the measure of the arc \( AB \) that it intercepts is:
\[ \text{m} \overset{\frown}{AB} = 2 \times 101^\circ = 202^\circ \]
Therefore, the correct answer is:
- **202°**
**Answer:**
1. *202°*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F32100c6a-eba5-49dd-834d-e3d8e02c9e26%2F6975f19f-102b-4aa7-a109-38b14eea0070%2Fhbunvja_processed.png&w=3840&q=75)
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