AB is a chord in a circle with centre O. AB is produced to C such that BC is equal to the radius of the circle. C is joined to O and produced to meet the circle at D. If `angle ACD = 32^@`, then the measure of `angle AOD` is .......

Options

A. `48^@`
B. `96^@`
C. `108^@`
D. `80^@`

Answer

To solve the problem step by step, we will analyze the given information and apply the properties of circles and angles.

### Step 1: Understand the Circle and Given Angles
We have a circle with center O, a chord AB, and a point C such that BC is equal to the radius of the circle. We are given that angle ACD = 32 degrees.

### Step 2: Identify the Triangle
Since BC is equal to the radius (let's denote the radius as r), we know that OB (the radius from O to point B) is also equal to r. Therefore, triangle OBC is isosceles with OB = OC = r.

### Step 3: Apply the Isosceles Triangle Property
In triangle OBC, since OB = OC, the angles opposite these sides are equal. Therefore, angle OBC = angle OCB. Let's denote these angles as x.

### Step 4: Use the Angle Sum Property
The sum of angles in triangle OBC is 180 degrees:
\[ \angle OBC + \angle OCB + \angle BOC = 180^\circ \]
Substituting the known values:
\[ x + x + \angle BOC = 180^\circ \]
This simplifies to:
\[ 2x + \angle BOC = 180^\circ \]
Thus,
\[ \angle BOC = 180^\circ - 2x \]

### Step 5: Relate Angle ACD to Angle BOC
We know that angle ACD = 32 degrees. By the properties of angles in circles, angle ACD is equal to angle BOC (since they subtend the same arc AD). Therefore:
\[ \angle BOC = 32^\circ \]

### Step 6: Substitute Back to Find x
Now we can substitute back into our equation:
\[ 180^\circ - 2x = 32^\circ \]
Rearranging gives:
\[ 2x = 180^\circ - 32^\circ \]
\[ 2x = 148^\circ \]
Thus,
\[ x = 74^\circ \]

### Step 7: Find Angle AOD
Now, angle AOD is the angle at the center of the circle that subtends the same arc AD as angle ACD. By the inscribed angle theorem, angle AOD is twice angle ACD:
\[ \angle AOD = 2 \times \angle ACD = 2 \times 32^\circ = 64^\circ \]

### Step 8: Final Calculation
We have found that angle AOD measures 64 degrees.

### Conclusion
The measure of angle AOD is **64 degrees**.