Prove that tangent drawn from external point are equal

Asked by admin @ 16/08/2021 in Math viewed by 325 People

Prove that the length of tangents drawn from an external point to a circle are equal

Answered by admin @ 16/08/2021

Let two tangent PT and QT are drawn to circle of centre O as shown in figure.
Both the given tangents PT and QT touch to the circle at P and Q respectively.

We have to proof : length of PT = length of QT
Construction :- draw a line segment ,from centre O to external point T { touching point of two tangents } .

Now ∆POT and ∆QOT
We know, tangent makes right angle with radius of circle.
Here, PO and QO are radii . So, ∠OPT = ∠OQT = 90°
Now, it is clear that both the triangles ∆POT and QOT are right angled triangle.
nd a common hypotenuse OT of these [ as shown in figure ]

Now, come to the concept ,
∆POT and ∆QOT
∠OPT = OQT = 90°
Common hypotenuse OT
And OP = OQ [ OP and OQ are radii]
So, R - H - S rule of similarity
∆POT ~ ∆QOT
Hence, OP/OQ = PT/QT = OT/OT
PT/QT = 1
PT = QT [ hence proved]

Prove that tangent drawn from external point are equal

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