If angle between two tangents drawn from a point P to a circle of radius a and center O is 90°, then


True


Let us consider a circle with center O and tangents PT and PR and angle between them is 90° and radius of circle is a


To show :


Proof :


In OTP and ORP


TO = OR [ radii of same circle]


OP = OP [ common ]


TP = PR [ tangents through an external point to a circle are equal]


OTP ORP [ By Side Side Side Criterion ]


TPO = OPR [Corresponding parts of congruent triangles are equal ] [1]


Now, TPR = 90° [Given]


TPO + OPR = 90°


TPO + TPO = 90° [By 1]


TP0 = 45°


Now, OT TP [ As tangent at any point on the circle is perpendicular to the radius through point of contact]


OTP = 90°


So POT is a right-angled triangle


And we know that,



So,


[As OT is radius and equal to a]




Hence Proved !


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