ABCDEFGH is a regualr octagon inscribedin a circle with centre at 0. The ratio of `angle OAB` to `angle AOB` is equal to :

Options

A. `4:3`
B. `3:1`
C. `3:2`
D. `8:3`

Answer

To find the ratio of angle \( OAB \) to angle \( AOB \) in a regular octagon inscribed in a circle, we can follow these steps:

### Step 1: Understand the Angles
In a regular octagon inscribed in a circle, the center of the circle is denoted as point \( O \), and points \( A \) and \( B \) are two consecutive vertices of the octagon.

### Step 2: Calculate Angle \( AOB \)
Since the octagon has 8 equal sides, the angle \( AOB \) can be calculated by dividing the total angle around point \( O \) (which is \( 360^\circ \)) by the number of sides (8):
\[ \text{Angle } AOB = \frac{360^\circ}{8} = 45^\circ \]

### Step 3: Analyze Triangle \( AOB \)
In triangle \( AOB \):
- \( AO \) and \( BO \) are the radii of the circle, hence they are equal.
- Therefore, triangle \( AOB \) is an isosceles triangle.

### Step 4: Use the Isosceles Triangle Property
Let \( \angle OAB = x \). Since \( AO = BO \), we have:
\[ \angle OAB = \angle OBA = x \]
Using the triangle sum property:
\[ \angle AOB + \angle OAB + \angle OBA = 180^\circ \]
Substituting the known values:
\[ 45^\circ + x + x = 180^\circ \]
This simplifies to:
\[ 2x + 45^\circ = 180^\circ \]
\[ 2x = 180^\circ - 45^\circ = 135^\circ \]
\[ x = \frac{135^\circ}{2} = 67.5^\circ \]

### Step 5: Calculate the Ratio
Now, we have:
- \( \angle OAB = 67.5^\circ \)
- \( \angle AOB = 45^\circ \)

The ratio of \( \angle OAB \) to \( \angle AOB \) is:
\[ \text{Ratio} = \frac{\angle OAB}{\angle AOB} = \frac{67.5^\circ}{45^\circ} = \frac{67.5}{45} = \frac{3}{2} \]

### Final Answer
Thus, the ratio of \( \angle OAB \) to \( \angle AOB \) is:
\[ \frac{3}{2} \]