Question

In the given figure, O is the centre of the circle. PQ and PR are tangents. Show that the quadrilateral PQOR is cyclic.

Hint:

A quadrilateral is cyclic if and only if its opposite angles sum to \( 180^\circ \).
Whenever you see tangents and radii meeting, identify the \( 90^\circ \) angles immediately, as they often lead directly to supplementary angle proofs.

Answer 1

Step 1: Understanding the Question:
The problem requires us to prove that the quadrilateral \( PQOR \), formed by two tangents \( PQ \) and \( PR \) from an external point \( P \) to a circle centered at \( O \) and the respective radii \( OQ \) and \( OR \), is a cyclic quadrilateral.
A cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle, which means its opposite angles are supplementary.

Step 2: Key Formula or Approach:
1. A key property of tangents states that the radius of a circle is perpendicular to the tangent at the point of contact.
2. Therefore, the angles formed between the radii and the tangents at points \( Q \) and \( R \) are right angles:
\[ \angle OQP = 90^\circ \quad \text{and} \quad \angle ORP = 90^\circ \]
3. To prove that a quadrilateral is cyclic, we must show that the sum of either pair of opposite angles is \( 180^\circ \).

Step 3: Detailed Explanation:
1. Let \( O \) be the center of the circle. \( PQ \) and \( PR \) are the tangents from an external point \( P \) touching the circle at \( Q \) and \( R \).
2. Since the radius is perpendicular to the tangent at the point of contact, we have:
\[ \angle OQP = 90^\circ \]
\[ \angle ORP = 90^\circ \]
3. Consider the quadrilateral \( PQOR \). The sum of all four interior angles of any quadrilateral is \( 360^\circ \):
\[ \angle QPR + \angle OQP + \angle QOR + \angle ORP = 360^\circ \]
4. Substitute the known perpendicular angles into this sum:
\[ \angle QPR + 90^\circ + \angle QOR + 90^\circ = 360^\circ \]
\[ \angle QPR + \angle QOR + 180^\circ = 360^\circ \]
5. Simplify the equation by subtracting \( 180^\circ \) from both sides:
\[ \angle QPR + \angle QOR = 180^\circ \]
6. Now, examine the sum of the opposite angles:
First pair of opposite angles: \( \angle OQP + \angle ORP = 90^\circ + 90^\circ = 180^\circ \).
Second pair of opposite angles: \( \angle QPR + \angle QOR = 180^\circ \).
7. Since the opposite angles of quadrilateral \( PQOR \) are supplementary, the quadrilateral is cyclic.

Step 4: Final Answer:
Hence, the quadrilateral \( PQOR \) is proven to be a cyclic quadrilateral.