A chord of a circle is equal to its radius. A tangent is drawn to the circle at an extremity of the chord. The angle between the tangent and the chord is

Options

A. `30^@`
B. `45^@`
C. `60^@`
D. `75^@`

Answer

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

### Step 1: Draw the Diagram
Draw a circle with center O. Mark a point A on the circle and draw a chord AB such that the length of the chord AB is equal to the radius OA.

**Hint:** Remember that the radius is the distance from the center of the circle to any point on the circle.

### Step 2: Identify Points and Tangent
Mark point B on the circle and draw a tangent line at point B. Let the tangent line extend to point D.

**Hint:** A tangent to a circle is a straight line that touches the circle at exactly one point.

### Step 3: Analyze the Triangle OAB
Since the chord AB is equal to the radius OA, we have OA = OB = AB. This means triangle OAB is an equilateral triangle.

**Hint:** In an equilateral triangle, all sides are equal, and all angles are equal to 60 degrees.

### Step 4: Calculate Angles in Triangle OAB
Since triangle OAB is equilateral, each angle (angle AOB, angle OAB, angle OBA) is equal to 60 degrees.

**Hint:** The sum of angles in any triangle is always 180 degrees.

### Step 5: Determine Angle DBC
Since DBC is a straight line, angle DBC = 180 degrees. We can express angle DBC as the sum of angles ABD, ABO, and OBC.

**Hint:** A straight line measures 180 degrees.

### Step 6: Substitute Known Angles
We know:
- Angle ABO = 60 degrees (from the equilateral triangle)
- Angle OBC = 90 degrees (because it is the angle between the radius and the tangent)

Now substitute these values into the equation:
\[ \text{Angle DBC} = \text{Angle ABD} + \text{Angle ABO} + \text{Angle OBC} \]
\[ 180 = \text{Angle ABD} + 60 + 90 \]

**Hint:** Use the known angles to simplify the equation.

### Step 7: Solve for Angle ABD
Now, simplify the equation:
\[ 180 = \text{Angle ABD} + 150 \]
\[ \text{Angle ABD} = 180 - 150 \]
\[ \text{Angle ABD} = 30 \text{ degrees} \]

**Hint:** Subtract the sum of the known angles from 180 degrees to find the unknown angle.

### Conclusion
The angle between the tangent (DB) and the chord (AB) is 30 degrees.

**Final Answer:** 30 degrees