Question

Prove that the angle between the two tangents drawn from an external point to a con
is supplementary to the angle subtended by the line-segment joining the
contact at the centre.
point​

Answer 1

Answer:

ANSWER

Draw a circle with center O and take a external point P. PA and PB are the tangents.

As radius of the circle is perpendicular to the tangent.

OA⊥PA

Similarly OB⊥PB

∠OBP=90

o

∠OAP=90

o

In Quadrilateral OAPB, sum of all interior angles =360

o

⇒∠OAP+∠OBP+∠BOA+∠APB=360

o

⇒90

o

+90

o

+∠BOA+∠APB=360

o

∠BOA+∠APB=180

o

It proves the angle between the two tangents drawn from an external point to a circle supplementary to the angle subtented by the line segment

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Answer 2

Step-by-step explanation:

LET PA & PB ARE THE TANGENTS DRAWN FROM AN EXTERNAL POINT TO THE CIRCLE WITH CENTER O JOIN OA AND OB.

TO PROVE :

ANGLE APB + ANGLE BOA = 180°

PROOF:

OA PERPENDICULAR TO PA [ RADIUS AT POINT OF CONTACT TO THE CIRCLE]

THEREFORE ANGLE OAP ≈90°

IN QUADRILATERAL OAPB,

ANGLE OAP + ANGLE APB + ANGLE PBO ± ANGLE BOA ≈ 360° [ SUM OF INTERIOR ANGLES]

»90°+ANGLE APB + 90°+ANGLE BOA =360°

»ANGLE APB + ANGLE BOA =180°

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