O is the circumcentre of the triangle ABC and `angle BAC = 85^@`, `angle BCA = 75^@` , then the value of `angle OAC` is

Options

A. `55^@`
B. `150^@`
C. `20^@`
D. `70^@`

Answer

To solve the problem, we need to find the value of angle OAC in triangle ABC, where O is the circumcenter. We are given the following angles:

- Angle BAC = 85°
- Angle BCA = 75°

Let's go through the solution step by step:

### Step 1: Find angle ABC
In any triangle, the sum of the angles is always 180°. Therefore, we can find angle ABC using the formula:

\[ \text{Angle ABC} = 180° - \text{Angle BAC} - \text{Angle BCA} \]

Substituting the given values:

\[ \text{Angle ABC} = 180° - 85° - 75° \]
\[ \text{Angle ABC} = 180° - 160° = 20° \]

### Step 2: Relate angle ABC to angle AOC
Since O is the circumcenter, angle AOC is twice the angle ABC (because the angle at the center is twice the angle at the circumference subtended by the same arc). Therefore:

\[ \text{Angle AOC} = 2 \times \text{Angle ABC} = 2 \times 20° = 40° \]

### Step 3: Analyze triangle AOC
In triangle AOC, we know:

- Angle AOC = 40°
- OA = OC (radii of the circumcircle)

Since OA = OC, triangle AOC is isosceles. Therefore, the angles OAC and OCA are equal. Let's denote these angles as y.

### Step 4: Set up the equation for triangle AOC
The sum of angles in triangle AOC is 180°. Thus, we have:

\[ \text{Angle OAC} + \text{Angle AOC} + \text{Angle OCA} = 180° \]
\[ y + 40° + y = 180° \]
\[ 2y + 40° = 180° \]

### Step 5: Solve for y
Now, we can solve for y:

\[ 2y = 180° - 40° \]
\[ 2y = 140° \]
\[ y = \frac{140°}{2} = 70° \]

### Conclusion
Thus, the value of angle OAC is:

\[ \text{Angle OAC} = 70° \]

### Final Answer
The value of angle OAC is **70°**.

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