Question

The perpendicular at the point of the tangent to a circle passes through the center of the circle.

• A. True
• B. False

Answer 1

The correct option is A True

Let us assume a circle with the center $O$ and $AB$ be the tangent intersecting circle at point $P.$

We know that,

Tangent of circle is perpendicular to radius at point of contact.

Hence, $OP\perp AB$

So, $\angle OPB={90}^o$ ------ ( 1 )

Now lets assume some point $X,$

Such that $XP\perp AB$

Hence, $\angle XPB={90}^o$ ------ ( 2 )

From ( 1 ) and ( 2 )

$\Rightarrow$ $\angle OPB=\angle XPB={90}^o$

Which is possible only if line $XP$ passes through $O.$

$\Rightarrow$ Hence, perpendicular to tangent passes through centers.