Question

In the given figure, PA is a tangent from an external point P to a circle with centre O. If $\angle POB = 125^\circ$, then $\angle APO$ is equal to :

• A. $25^\circ$
• B. $65^\circ$
• C. $90^\circ$
• D. $35^\circ$

Hint:

An alternative shortcut is using the exterior angle theorem of triangles.
In $\Delta OAP$, the exterior angle $\angle POB$ at vertex $O$ is equal to the sum of the two interior opposite angles:
\[ \angle POB = \angle OAP + \angle APO \]
\[ 125^\circ = 90^\circ + \angle APO \implies \angle APO = 35^\circ \]
This approach avoids calculating the linear pair and directly yields the answer.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given a circle with centre $O$. $PA$ is a tangent to the circle from an external point $P$.
A line segment $AB$ is a diameter of the circle, making $A, O, B$ collinear (a straight line).
The exterior angle is given as $\angle POB = 125^\circ$. We need to find the angle $\angle APO$.

Step 2: Key Formula or Approach:
We will use the following properties:
1. Angles on a straight line (linear pair) sum up to $180^\circ$:
\[ \angle AOP + \angle POB = 180^\circ \]
2. A radius is perpendicular to the tangent at the point of contact:
\[ \angle OAP = 90^\circ \]
3. The sum of the angles in a triangle is $180^\circ$. In $\Delta OAP$:
\[ \angle APO + \angle OAP + \angle AOP = 180^\circ \]

Step 3: Detailed Explanation:

• 1. Determine the interior angle $\angle AOP$ using the linear pair property with the given angle $\angle POB = 125^\circ$:
\[ \angle AOP + 125^\circ = 180^\circ \]
\[ \angle AOP = 180^\circ - 125^\circ = 55^\circ \]

• 2. State the relationship between the radius $OA$ and the tangent $PA$ at the point of contact $A$:
\[ \angle OAP = 90^\circ \]

• 3. Use the angle sum property in $\Delta OAP$:
\[ \angle APO + \angle OAP + \angle AOP = 180^\circ \]

• 4. Substitute the values of $\angle OAP = 90^\circ$ and $\angle AOP = 55^\circ$:
\[ \angle APO + 90^\circ + 55^\circ = 180^\circ \]

• 5. Solve for $\angle APO$:
\[ \angle APO + 145^\circ = 180^\circ \]
\[ \angle APO = 180^\circ - 145^\circ = 35^\circ \]


Step 4: Final Answer:
The angle $\angle APO$ is equal to $35^\circ$, which corresponds to option (D).