Question

Proof of the theorem tangent segments drawn from an external point to a circle are congruent​

Answer 1

Answer:

angle q=angle r =90°[point of contact ]

oq=or (radii )

po=po (common)

by rhs congruency

pqo=pro

by cpct

pq=pr

Answer 2

The theorem tangent segments drawn from an external point to a circle are congruent​.

• This theorem is also called as the "hat theorem" as the circle with center O looks like wearing a hat named ABC.
• Given,
• A circle with center O.
• AB and BC are the tangents to the circle from the same external point B.
• Construction:
• Join OB.
• Proof:
• Consider ΔBAO and ΔBCO
• OB = OB (both share same hypothesis)
• OA = OC (radii of the same circle)
• ∠BAO = ∠BCO (both measure 90°)

• Therefore,
• from SAS theorem and CPCTC (corresponding parts of congruent triangles are congruent)
• we have proved that,
• the theorem tangent segments drawn from an external point to a circle are congruent​.