To prove that tangents PQ and PR are equal in length without any extra constructions, we can utilize the properties of tangents to a circle.

Tangents PQ and PR to a circle with center O.

To prove: PQ = PR.

Proof:

OQ is common to both triangles:*

OQ is the radius of the circle, and it is common to both triangles.

3. OR is common to both triangles:

OR is the radius of the circle, and it is common to both triangles.

4. QOR = ∠PQR (both are right angles):

The common side OQ and OR are shared, and the right angles are equal.

5. By the Right-Angle Hypotenuse-Leg Congruence Theorem:

• Triangles QOR and PQR are congruent.

5. Therefore, corresponding sides are equal: