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Circles

A circle has important elements such as chords, secants, and tangents. Go through the given video to understand these concepts.

The theorem about tangents states that:

A tangent at any point of a circle is perpendicular to the radius through the point of contact.

In the above figure O is the centre of circle, line l is the tangent and P is point of contact.

l ⊥ OP

Proof:

It is given that O is the centre of the circle, l is the tangent to this circle and P is the point of contact.

Let us assume l is not perpendicular to the radius of the circle.

In this case, let us draw perpendicular OA to tangent l. Thus, point A is distinct from point P.

Let B be any point on tangent such that BAP is a line and BA = AP.

Now, in ΔOAB and ΔOAP, we have

OA = OA        (Common side)

∠OAB = ∠OAP        (OA ⊥ tangent l)

BA = AP        (By construction)

∴ ΔOABΔOAP

∴ OB = OP        (By CPCT)

Since OB = OP, point B also lies on the circle.

Also, point B is different from point P.

Thus, tangent l touches the circle at two distinct points. This contradicts the ...

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