I drew a circle having PR as a diameter. I draw a tangent at tangent at the point P and a point S is taken on the tangent of the circle in such a way that PR = PS. If RS intersects the circle at the point T. Let us prove that ST = RT = PT.

Theory.

Angle sum property of triangle is 180°

if 2 sides of triangle are equal then their corresponding angles will also be equal

Solution.

In Δ PRS

PS = PR

PSR = PRS

RPS = 90° (Radius of circle from point of contact of tangent is 90° )

PSR + PRS + RPS = 180°

2PSR = 180° - 90°

PSR = 45°

PSR = PRS = 45°

In Δ PRT

PTR = 90° (3rd point of triangle on circumference of semicircle is always 90° )

PRT = PRS = 45°

TPR + PRT + PTR = 180°

TPR = 180° - 135°

= 45°

PRT = TPR

RT = TP (isosceles triangle property)………1

In Δ PTS

RPS = TPS + TPR = 90°

TPS + 45° = 90°

TPS = 45°

PST = PSR = 45° (proved above)

PST = TPS

PT = ST (isosceles triangle property)………2

Joining 1 and 2

We get ;

PT = ST = RT

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