Question

5. Prove that the perpendicular at the point of contact to the tangent to a circle passes
through the centre
​

Answer 1

Step-by-step explanation:

O is the centre of the given circle.

A tangent PR has been drawn touching the circle at point P.

Draw QP ⊥ RP at point P, such that point Q lies on the circle.OPR = 90° (radius ⊥ tangent)

Also, ∠QPR = 90° (Given)

∴ ∠OPR = ∠QPR

Now, above case is possible only when centre O lies on the line QP.

Hence, perpendicular at the point of contact to the tangent to a circle passes through the centre of the circle.

Answer 2

$\mathfrak{ \huge{ \red{ \underline{QUESTɪOɴ : }}}}$

PʀOVE TʜAT TʜE PEʀPEɴDɪCUʟAʀ AT TʜE POɪɴT OF COɴTACT TO TʜE TAɴɢEɴT TO A CɪʀCʟE PASSES TʜʀOUɢʜ TʜE CEɴTʀE ʟET , O ɪS TʜE CEɴTʀE OF TʜE ɢɪVEɴ CɪʀCʟE .

$\mathfrak{ \huge{ \green{ \underline{ SOʟUTɪOɴ : }}}}$

A tangent PR has been drawn touching the circle at point P.

Draw QP ⊥ RP at point P, such that point Q lies on the circle.

∠OPR = 90° (radius ⊥ tangent)

Also, ∠QPR = 90° (Given)

∴ ∠OPR = ∠QPR

Now, the above case is possible only when centre O lies on the line QP.

Hence, perpendicular at the point of contact to the tangent to a circle passes through the centre of the circle.