# 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.

Byju's Answer Standard X Mathematics

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)

we have already assumed that

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.**

**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 contact

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

**Q.**

**Question 5**

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## 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

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## 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)

Last updated at Aug. 5, 2021 by Teachoo

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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 →

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### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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