Question

The lengths of the two tangents from an external point to a circle are _____.

• A. constant
• B. equal
• C. unequal
• D. not a constant

Hint:

Whenever two tangents are drawn from the same external point: \[ \text{Left Tangent} = \text{Right Tangent} \] This property is frequently used in geometry proofs and numerical problems.

Answer 1

The Correct Option is B

Concept: One of the most important properties of tangents to a circle is: \[ \text{Tangents drawn from the same external point to a circle are equal in length.} \] If:
• \(P\) is an external point,
• \(PA\) and \(PB\) are tangents touching the circle at points \(A\) and \(B\), then: \[ PA = PB \]

Step 1: Understand the geometric situation. Suppose:
• \(O\) is the center of the circle,
• \(P\) is a point outside the circle,
• \(PA\) and \(PB\) are tangents touching the circle at \(A\) and \(B\). The figure forms two triangles: \[ \triangle OPA \quad \text{and} \quad \triangle OPB \]

Step 2: Use properties of tangents and radii. A radius drawn to the point of contact is always perpendicular to the tangent. Therefore: \[ OA \perp PA \] and: \[ OB \perp PB \] Thus: \[ \angle OAP = \angle OBP = 90^\circ \]

Step 3: Compare the two triangles. In triangles \( \triangle OPA \) and \( \triangle OPB \):
• \(OA = OB\) (radii of the same circle)
• \(OP\) is common
• Both are right triangles Hence by RHS congruence: \[ \triangle OPA \cong \triangle OPB \]

Step 4: Conclude equality of tangent lengths. Since the triangles are congruent: \[ PA = PB \] Therefore, the lengths of tangents drawn from an external point are always equal. Final Answer: \[ \boxed{\text{equal}} \]