Question

prove that the tangent at any point of a circle is perpendicular to the radius through the point of the contact​

Answer 1

$\huge{\underline{\underline{\red{Answer}}}}$

$\mathbb{\boxed{\pink{NEED\:TO\:PROVE-}}}$

• Tangent at any point of a circle is perpendicular to radius through point of contact.

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$\large{\bf{\underline{\red{SOLUTION-}}}}$

Consider

▪ a circle with centre O.

▪CD be the tangent

▪Let OA and OE be the radius

▪OP be a line joining the centre and the tangent

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Now, OA= OE----ii (radius of same circle)

From the figure,

OP= OE+EP

So, we can say

$\bold{\blue{\boxed{OP>OE---i}}}$

from i and ii we can say,

OP > OA (as OA=OE)

Thus OA(the radius) is the shortest point on The tangent, CD.------A

We know the perpendicular to any point from a line is the shortest distance between them.-------B

considering A and B we can say,

Tangent to any point is perpendicular to the radius at point of contact.$\boxed{\boxed{\green{PROVED}}}$

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$\bold{\pink{\boxed{Knowledge\:Cell}}}$

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How to prove that a perpendicular to a point from anotger is the shortest distance between them?▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪▪

$\bf{\boxed{\red{SOLUTION-}}}$

Consider a right angled triangle PQR,

PQ= perpendicular

PR= Hypotenuse.

We know,

${pr}^{2} = {pq}^{2} + {qr}^{2}$

From this we can say,

${pr}^{2} > {pq}^{2} \\ \implies \: pr > pq$

Thus the perpendicular is the shortest distance between any two points.

Answer 2

$\huge\underline\mathbb{SOLUTION:-}$

Given:

• A circle c (O,R) and a tangent L at point A.

Need To Prove:

• OA $\perp$L

Construction:

• Take a point B, other than A, on the tangent L.
• Join OB.
• Suppose OB meets the circle C.

Proof:

• We know that, among all line segment joining the point O to a point on L, the perpendicular is shortest to L.

OA = OC (Radius of the circle same)

Now, OB = OC + BC

Therefore,

OB > OC (OC being radius and B any point on tangent)

$\implies$OB > OA

$\implies$ OA < OB

B is the arbitrary point on the tangent.

Thus, OA is shorter than any other line segment joining O to any point on tangent.

Here,

• OA $\perp$L

Shortest distance of a point from a given line is the perpendicular distance from that line.

Hence, the tangent at any point of circle is perpendicular to the radius.