A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc

Options

A. `150^@`
B. `180^@`
C. `160^@`
D. `170^@`

Explanation

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Understand the Given Information
We have a circle with center O, and a chord AB that is equal in length to the radius of the circle. This means:
- AB = OA = OB (where OA and OB are the radii of the circle).

### Step 2: Identify the Triangle Formed
Since AB is equal to the radius, triangle AOB is an equilateral triangle. In an equilateral triangle, all sides are equal and all angles are equal.

### Step 3: Calculate the Angles of Triangle AOB
In an equilateral triangle, each angle measures:
\[ \text{Angle AOB} = \text{Angle OAB} = \text{Angle OBA} = 60^\circ \]

### Step 4: Relate Angle AOB to Angles on the Arcs
The angle AOB subtended by the chord AB at the center O is related to the angles subtended at points on the circle. Specifically, the angle subtended at any point on the circle (like point D on the minor arc) is half of the angle at the center:
\[ \text{Angle ADB} = \frac{1}{2} \times \text{Angle AOB} = \frac{1}{2} \times 60^\circ = 30^\circ \]

### Step 5: Find the Angle on the Major Arc
To find the angle subtended by the chord AB at a point on the major arc (like point C), we can use the property of cyclic quadrilaterals. The angles subtended by the same chord at points on the circle add up to 180 degrees:
\[ \text{Angle ACB} + \text{Angle ADB} = 180^\circ \]
We already calculated angle ADB as 30 degrees. Therefore:
\[ \text{Angle ACB} = 180^\circ - \text{Angle ADB} = 180^\circ - 30^\circ = 150^\circ \]

### Final Results
- The angle subtended by the chord AB at point D on the minor arc is **30 degrees**.
- The angle subtended by the chord AB at point C on the major arc is **150 degrees**.