AB and AC are tangents to a circle with centre O. A is the external point of the circle. The line AO intersect the chord BC at D. The measure of the `/_BDO `is

Options

A. `60^@`
B. `90^@`
C. `45^@`
D. `75^@`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Configuration
We have a circle with center O and two tangents AB and AC drawn from an external point A. The line AO intersects the chord BC at point D. We need to find the measure of angle BDO.

### Step 2: Identify Key Properties
Since AB and AC are tangents to the circle from point A, we know that:
- OA is the radius to the point of tangency, making angles OAB and OAC both equal to 90 degrees.
- The lengths of the tangents from a point outside the circle to the points of tangency are equal, i.e., AB = AC.

### Step 3: Establish Congruent Triangles
We can observe two triangles:
- Triangle AOB
- Triangle AOC

Since:
- OA = OA (common side)
- AB = AC (tangent lengths)
- Angle OAB = Angle OAC = 90 degrees

By the criteria of congruence (RHS), we can conclude that:
- Triangle AOB is congruent to Triangle AOC.

### Step 4: Analyze Angles in Congruent Triangles
From the congruence of triangles AOB and AOC, we have:
- Angle AOB = Angle AOC.

### Step 5: Consider Triangles ABD and ACD
Now, we will look at triangles ABD and ACD:
- AD is a common side.
- AB = AC (from the tangents).
- Angle ADB = Angle ADC (as both angles subtend the same arc BC).

Thus, triangles ABD and ACD are also congruent by the criteria AAS (Angle-Angle-Side).

### Step 6: Conclude the Measure of Angle BDO
From the congruence of triangles ABD and ACD, we have:
- Angle ADB = Angle ADC.

Let’s denote these angles as θ. Therefore:
- Angle ADB + Angle ADC = 180 degrees (since they are supplementary).

This gives us:
- 2θ = 180 degrees
- θ = 90 degrees.

Thus, angle BDO, which is equal to angle ADB, is also 90 degrees.

### Final Answer
The measure of angle BDO is **90 degrees**.

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