Question

the tangents at any point of a circle is perpendicular to the radius through the point of contact ​

Answer 1

Answer refer to Attachment

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Answer 2

Qᴜᴇsᴛɪᴏɴ :

➥ The tangents at any point of a circle is perpendicular to the radius through the point of contact.

Pʀᴏᴠᴇᴅ :

➥ OP ⊥ AB

Gɪᴠᴇɴ :

➤ A circle with center O and a tangent AB at a point P of the circle.

Tᴏ Pʀᴏᴠᴇ :

➤ OP ⊥ AB ?

Cᴏɴꜱᴛʀᴜᴄᴛɪᴏɴ :

➤ Take a point Q, other than P, on AB. Join OQ.

Pʀᴏᴏꜰ :

Q is a point on the tangent AB, other than the point of contact P.

∴ Q lies outside the circle

Let OQ intersect the circle at R

Then, OR < OQ ⠀ [a part is less than the whole] ...❶

But, OP = OR ⠀ [radii of the same circle whole] ...❷

∴ OR < OQ ⠀ [from equ ❶ and ❷]

Thus, OP is shorter than any other line segment joining O to any point point of AB, other than P

In other words, OP is the shortest distance between the point O and the line AB

But, the shortest distance between a point and a line is the perpendicular distance

$:\implies \underline{\overline{\boxed{\purple{\bf{\:\:\therefore OP \perp AB \:\:}}}}}$【 PROVED 】