Question

o is the centre of circle.PT and PQ are tangents to the circle from an external point P.if angle TPQ=70°,find angleTRQ

Answer 1

Please see attached document

Hope it helps you

Answer 2

Given,

O is the center of a circle.

PT and PQ are two tangents to the circle from an external point P

Angle TPQ=70°

To find,

The measure of angle TRQ.

Solution,

We can simply solve this mathematical problem using the following process:

As per geometry of a circle;

A tangent to a circle is perpendicular to the radius of the circle at that point.

{Statement-1}

Sum of all the angles of any quadrilateral = 180°

{Statement-2}

In any circle, a chord subtends an angle that is double the angle it subtends at any other point on the major arc of the circle.

{Statement-3}

Now, according to the question;

The two tangents PT and PQ are perpendicular to OT and OQ, respectively.

(according to statement-1)

=> angle OTP = angle OQP = 90°

(Equation-1)

Also, according to the statement-2;

Sum of all the angles of the quadrilateral OTPQ = 180°

=> angle TOQ + angle OTP + angle OQP + angle TPQ = 360°

=> angle TOQ + 90° + 90° + 70° = 360°

(according to equation-1)

=> angle TOQ = 110°

Now, according to statement-3;

angle TOQ = 2 × angle TRQ

=> angle TRQ = angle TOQ / 2 = 110°/2

=> angle TRQ = 55°

Hence, the measure of angle TRQ is equal to 55°.