In the diagram, PQ and QR are tangents to the circle with centre O, at P and R respectively. Find the measure of x.

To solve the problem, we can use the property of tangents to a circle. The lengths of tangents drawn from an external point to a circle are equal.

Step 1: Identify Variables and Relationships

Let:

• $PQ$ and $QR$ be the tangents from point $Q$ to points $P$ and $R$ on the circle.
• The center of the circle is $O$.

Step 2: Apply Tangent Properties

Since $PQ$ and $QR$ are tangents:

• $PQ = QR$.

Step 3: Set up Equations

Assuming the measure of angle $PQR$ equals to $x$, we know:

• The angle between a tangent and a radius drawn to the point of contact is $90^\circ$. Therefore:
  • $\angle OPQ = 90^\circ$
  • $\angle OQR = 90^\circ$

Step 4: Use Angle Relationships

From triangle $PQR$:

• The sum of angles in triangle $PQR$ is $180^\circ$: $\angle PQR + \angle OPQ + \angle OQR = 180^\circ$ Substituting the known angles: $x + 90^\circ + 90^\circ = 180^\circ$

Step 5: Solve for x

$x + 180^\circ = 180^\circ$ $x = 0^\circ$

Final Answer

Thus, the measure of $x$ is $\mathbf{0^\circ}$.