O is a point on line AB. Rays OC and OD are drawn on the same side of AB such that OC ⊥ OD, ∠AOC=x and ∠BOD=3x. Then, ∠BOD is equal to :

The correct option is $67\frac{1°}{2}$.

Step-by-step Explanation:

1. Understand the Geometric Setup:
We are given a straight line AB with point O lying on it. Rays OC and OD are drawn on the same side of line AB.
Since AB is a straight line, the total angle formed on one side of line AB at point O is a straight angle, which equals 180°.

2. Identify the Given Angles:
- Ray OC is perpendicular to Ray OD (OC ⊥ OD), which means:

$\angle COD=90°$

- $\angle AOC=x$

- $\angle BOD=3x$

3. Formulate the Equation:
The sum of all adjacent angles on the straight line AB at point O is 180°:

$\angle AOC+\angle COD+\angle BOD=180°$

Substitute the given values into the equation:

$x+90°+3x=180°$

4. Solve for x:
Combine like terms:

$4x+90°=180°$

Subtract 90° from both sides:

$4x=90°$

Divide by 4:

$x=\frac{90°}{4}=22.5°=22\frac{1°}{2}$

5. Calculate ∠BOD:
Now, find the value of ∠BOD which is defined as 3x:

$\angle BOD=3x=3\times 22.5°=67.5°=67\frac{1°}{2}$

Therefore, ∠BOD is equal to $67\frac{1°}{2}$.