Prove that the lengths of tangents drawn from an external point to a circle are equal.

Step 1: Draw the Diagram

Let's start by visualizing the problem. We have a circle with its center at point O. From an external point P, two tangents, PQ and PR, are drawn to the circle. Q and R are the points where the tangents touch the circle.

Step 2: Construct Radii and Join Center to External Point

To prove the equality of the tangent lengths, we need to create some triangles. We will join the center O to the points of tangency Q and R, forming radii OQ and OR. We will also join the center O to the external point P, forming the line segment OP.

Step 3: Identify Right Angles

A fundamental property of circles states that the radius drawn to the point of tangency is perpendicular to the tangent. Therefore, OQ is perpendicular to PQ, making angle OQP a right angle. Similarly, OR is perpendicular to PR, making angle ORP a right angle.

Step 4: Prove Triangle Congruence

Now consider the two right-angled triangles, OQP and ORP. We know that OQ and OR are radii of the same circle, so they are equal in length. Both triangles have a right angle (at Q and R). The side OP is common to both triangles. By the RHS (Right angle-Hypotenuse-Side) congruence rule, these two triangles are congruent.

Step 5: Conclude Equality of Tangent Lengths

Since triangles OQP and ORP are congruent, their corresponding parts are equal (CPCTC - Corresponding Parts of Congruent Triangles are Congruent). Therefore, the length of tangent PQ is equal to the length of tangent PR. This completes the proof.

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