Question

If TP and TQ are two tangents to a circle with centre O from an external point T so that $\angle POQ = 120^\circ$, then $\angle PTQ$ is equal to :

• A. $60^\circ$
• B. $70^\circ$
• C. $80^\circ$
• D. $90^\circ$

Hint:

The angle between two tangents drawn from an external point and the angle subtended by the line segment joining the points of contact at the centre are supplementary.
This means:
\[ \angle PTQ + \angle POQ = 180^\circ \]
Using this directly, we get:
\[ \angle PTQ = 180^\circ - 120^\circ = 60^\circ \]
This shortcut saves significant calculation time during exams.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The topic of this question is Circles, specifically the properties of tangents drawn from an external point.
We are given two tangents, $TP$ and $TQ$, drawn from an external point $T$ to a circle with centre $O$.
The angle subtended by the points of contact at the centre is $\angle POQ = 120^\circ$.
We need to calculate the angle between the two tangents, which is $\angle PTQ$.

Step 2: Key Formula or Approach:
We use two fundamental geometric theorems:

• Theorem 1: A tangent at any point of a circle is perpendicular to the radius through the point of contact. Therefore, $\angle OPT = 90^\circ$ and $\angle OQT = 90^\circ$.

• Theorem 2: The sum of the interior angles of a quadrilateral is $360^\circ$.

Using these, we can set up an angle sum equation for the quadrilateral $OPTQ$.

Step 3: Detailed Explanation:

• Let $O$ be the centre of the circle, and $T$ be the external point.

• $TP$ and $TQ$ are the tangents contacting the circle at points $P$ and $Q$ respectively.

• Since the radius is perpendicular to the tangent at the point of contact:
\[ \angle OPT = 90^\circ \]
\[ \angle OQT = 90^\circ \]

• Consider the quadrilateral $OPTQ$ formed by the points $O, P, T,$ and $Q$.

• The sum of all four interior angles in quadrilateral $OPTQ$ is:
\[ \angle PTQ + \angle OPT + \angle POQ + \angle OQT = 360^\circ \]

• Substitute the known values ($\angle OPT = 90^\circ$, $\angle OQT = 90^\circ$, and $\angle POQ = 120^\circ$) into the equation:
\[ \angle PTQ + 90^\circ + 120^\circ + 90^\circ = 360^\circ \]

• Simplify the sum of the angles:
\[ \angle PTQ + 300^\circ = 360^\circ \]

• Solve for $\angle PTQ$:
\[ \angle PTQ = 360^\circ - 300^\circ = 60^\circ \]


Step 4: Final Answer:
The angle $\angle PTQ$ is equal to $60^\circ$, which corresponds to option (A).