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# prove that the perpendicular at the point of contact to a tangent to a circle

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

Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre. Tangent Perpendicular to Radius at Point of Contact

Prove that th... Question

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

Open in App Solution

STEP :Proof Let us assume a circle with centre

O and let AB

be the tangent intersecting the circle at point

P .

Also let us assume a point

X such that XP is perpendicular to AB .

STEP 2 : Proving that

XP

passes through centre

O We know that

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

⇒OP⊥AB

(Tangent at any point of circle is perpendicular to the radius through point of contact)

So, ∠OPB=90° ...(1)

XP is perpendicular to AB ∠XPB=90° ...(2) Now from equation (1) and (2) ∠OPB=∠XPB=90°

This condition is possible only if line

XP passes through O . Since, XP

passes through centre

O .

Therefore, it is proved that the perpendicular at the point of contact to the tangent of a circle passes through the centre.

Suggest Corrections 42 SIMILAR QUESTIONS

Q.

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

Q.

State true or false:

For a tangent, the perpendicular line from the point of contact to the circle passes through the centre.

Q. Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contactQ. Prove that tangent drawn at any point of a circle is perpendicular to the radius through the point of contact.Q. Question 5

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

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स्रोत : byjus.com

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

Click here👆to get an answer to your question ✍️ Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre of the circle. Question

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

Medium Open in App Solution Verified by Toppr

Given a circle with center O and AB the tangent intersecting circle at point P

and prove that OP⊥AB

We know that tangent of the circle is perpendicular to radius at points of contact Hence

OP⊥AB So, ∠OPB=90 o ..........(i)

Now lets assume some point X

Such that XP⊥AN Hence ∠XPB=90 o .........(ii) From eq (i) & (ii) ∠OPB=∠XPB=90 o

Which is possible only if line XP passes though O

Hence perpendicular to tangent passes though centre 639 164

स्रोत : www.toppr.com

## Ex 10.2, 5

Ex 10.2,5 Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre. Given: Let us assume a circle with centre O & AB be the tangent intersecting circle at point P To prove: OP AB Proof: We know that Tangent of circle is perpendicular t Check sibling questions

## Ex 10.2, 5 - Chapter 10 Class 10 Circles (Term 2)

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### Transcript

Ex 10.2,5 Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre. Given: Let us assume a circle with centre O & AB be the tangent intersecting circle at point P To prove: OP AB Proof: We know that Tangent of circle is perpendicular to radius at point of contact Hence, OP AB So, OPB = 90 Now lets assume some point X , such that XP AB Hence, XPB = 90 From (1) and (2) OPB = XPB = 90 Which is possible only if line XP passes through O Hence , perpendicular to tangent passes through centre

Next: Ex 10.2, 6 → 