Question

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

Hint:

A neat diagram showing the circle, tangents, and radii is essential for full marks in geometry proofs.

Answer 1

Step 1: Understanding the Concept:
We use the properties of a circle and congruent triangles. The radius is perpendicular to the tangent at the point of contact.
Step 2: Detailed Explanation:
Given: A circle with centre O and two tangents PQ and PR drawn from an external point P.
To Prove: \( PQ = PR \).
Construction: Join OQ, OR and OP.
Proof:
In \( \triangle OQP \) and \( \triangle ORP \):
1. \( OQ = OR \) (Radii of the same circle).
2. \( \angle OQP = \angle ORP = 90^\circ \) (The tangent at any point of a circle is perpendicular to the radius through the point of contact).
3. \( OP = OP \) (Common side).
Therefore, \( \triangle OQP \cong \triangle ORP \) by the RHS (Right angle-Hypotenuse-Side) congruence rule.
By CPCT (Corresponding Parts of Congruent Triangles):
\[ PQ = PR \]
Step 3: Final Answer:
Lengths of tangents from an external point are equal. Hence Proved.