O is the circumcentre of a triangle ABC. If `angleBAC=85^(@) and angleBCA=75^(@)` then what is the value of `angleOAC`?

Options

A. `40^(@)`
B. `60^(@)`
C. `70^(@)`
D. `90^(@)`

Answer

To find the value of angle OAC in triangle ABC, where O is the circumcenter, we can follow these steps:

### Step 1: Identify the given angles
We are given:
- Angle BAC = 85°
- Angle BCA = 75°

### Step 2: Find the third angle ABC
Using the property that the sum of angles in a triangle is 180°, we can find angle ABC.

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

### Step 3: Relate angle AOC to angle ABC
Since O is the circumcenter, the angle at the circumcenter (angle AOC) is twice the angle at the circumference (angle ABC).

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

### Step 4: Use properties of isosceles triangles
In triangle OAC, since O is the circumcenter, we know that OA = OC (radii of the circumcircle). Therefore, triangle OAC is isosceles, which means:

\[ \text{Angle OAC} = \text{Angle OCA} \]

Let angle OAC = angle OCA = x.

### Step 5: Set up the equation for triangle OAC
The sum of the angles in triangle OAC is 180°:

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

### Step 6: Solve for x
Subtract 40° from both sides:

\[ 2x = 180° - 40° \]
\[ 2x = 140° \]

Now divide by 2:

\[ x = 70° \]

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

\[ \text{Angle OAC} = 70° \]

---