Question

please answer guys
prove that the length of tangents drawn from an external point to a circle are equal

Answer 1

Draw a circle, take a point outside it P

now join P with centre of cirrcle O

Draw 2 tangents to circle from point P

Mark points Q and R where tangent touches circle

Join P and Q with centre O

now, u have 2 triangles named PQO and PRO

in these triangles
Angle O = Angle R (90 degree each)
QO = RO (radius of same circle)
PO = PO ( common side)
Therefore by RHS , Triangle PQO and PRO are congurent

so, Tangent PR = PQ
Hence Proved

Answer 2

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]

hope it helps you dear..