A, B and C are three points on a circle with centre O. The tangent at C meets BA produced to T. If `/_ATC = 30^@` and `/_ACT = 48^@`, then what is the value of `/_AOB `?

Options

A. `78^@`
B. `96^@`
C. `102^@`
D. `108^@`

Answer

To find the value of angle \( \angle AOB \), we will follow these steps:

### Step 1: Identify the angles in triangle ACT
We know:
- \( \angle ATC = 30^\circ \)
- \( \angle ACT = 48^\circ \)

Using the triangle sum property, we can find \( \angle CAT \):
\[ \angle CAT = 180^\circ - \angle ATC - \angle ACT \]
Substituting the values:
\[ \angle CAT = 180^\circ - 30^\circ - 48^\circ = 102^\circ \]

### Step 2: Relate angles with the center O
Since \( OC \) is a radius and \( CT \) is a tangent at point C, we know that:
\[ \angle OCA = 90^\circ \]
Now, we can find \( \angle OAC \):
\[ \angle OAC = 90^\circ - \angle ACT = 90^\circ - 48^\circ = 42^\circ \]

### Step 3: Find angle AOB
In triangle \( OAC \), we have:
- \( \angle OAC = 42^\circ \)
- \( \angle OCA = 90^\circ \)

Using the triangle sum property again:
\[ \angle AOC = 180^\circ - \angle OAC - \angle OCA \]
Substituting the values:
\[ \angle AOC = 180^\circ - 42^\circ - 90^\circ = 48^\circ \]

### Step 4: Relate \( \angle AOB \) with \( \angle AOC \)
Since \( O \) is the center of the circle, the angle \( \angle AOB \) is twice the angle \( \angle ACB \) (the angle subtended by the same arc):
\[ \angle AOB = 2 \times \angle AOC = 2 \times 48^\circ = 96^\circ \]

### Final Answer
Thus, the value of \( \angle AOB \) is:
\[ \angle AOB = 96^\circ \]

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