`DeltaABC `is inscribed in a circle so that BC is diameter. The tangent at a point C intersects BA when produced at a point D. If `/_ABC = 36^@` then the value of `/_ADC `is

Options

A. `36^@`
B. `44^@`
C. `48^@`
D. `54^@`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We have triangle \( \Delta ABC \) inscribed in a circle where \( BC \) is the diameter. The tangent at point \( C \) intersects line \( BA \) when extended at point \( D \). We are given that \( \angle ABC = 36^\circ \).

### Step 2: Identify the angles in the triangle
Since \( BC \) is the diameter of the circle, by the inscribed angle theorem, we know that \( \angle BAC \) is a right angle (90 degrees). This is because any angle inscribed in a semicircle is a right angle.

### Step 3: Set up the equation for triangle \( ABC \)
In triangle \( ABC \), we have:
- \( \angle ABC = 36^\circ \)
- \( \angle BAC = 90^\circ \)
- Let \( \angle ACB = \theta \).

Using the triangle sum property, we know that the sum of angles in a triangle is \( 180^\circ \):
\[ \angle ABC + \angle BAC + \angle ACB = 180^\circ \]
Substituting the known values:
\[ 36^\circ + 90^\circ + \theta = 180^\circ \]

### Step 4: Solve for \( \theta \)
Now, we can solve for \( \theta \):
\[ 126^\circ + \theta = 180^\circ \]
\[ \theta = 180^\circ - 126^\circ = 54^\circ \]
Thus, \( \angle ACB = 54^\circ \).

### Step 5: Relate \( \angle ACB \) to \( \angle ADC \)
Since \( CD \) is a tangent to the circle at point \( C \), we know that the angle between the tangent and the radius at the point of tangency is \( 90^\circ \). Therefore, we have:
\[ \angle ACD = 90^\circ \]

### Step 6: Find \( \angle ADC \)
Now, we can find \( \angle ADC \) using the triangle \( ACD \):
\[ \angle ACD + \angle ACB + \angle ADC = 180^\circ \]
Substituting the known values:
\[ 90^\circ + 54^\circ + \angle ADC = 180^\circ \]
\[ 144^\circ + \angle ADC = 180^\circ \]
\[ \angle ADC = 180^\circ - 144^\circ = 36^\circ \]

### Final Answer
The value of \( \angle ADC \) is \( 54^\circ \).