Prove that the tangent and the radius through the point of contact of a circle are perpendicular to each other.

We need to prove that the tangent and the radius through the point of contact of a circle are perpendicular to each other. Given: A B is the tangent to the circle with centre O and P is the point of contact. O P is the radius of the circle. To prove: O P ⊥ A B Take any point Q on the tangent A B other than the point P . The point Q lies outside the circle. O Q > r , so O Q > O P , where r is the radius of the circle. So, we can say that O P is the shortest distance between point O and the line A B . We know that the shortest line segment drawn from a point to a given line is perpendicular to the line. Hence, O P ⊥ A B .

E M B I B E

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 4 - Theorems Regarding Tangent to a Circle>West Bengal Board-2019>Q 2

MEDIUM

10th West Bengal Board

IMPORTANT

Earn 100

If two circles touch each other externally then the point of contact will lie on the line segment joining the two centres -Prove it.

Important Questions on Theorems Regarding Tangent to a Circle

MEDIUM

10th West Bengal Board

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 4 - Theorems Regarding Tangent to a Circle>West Bengal Board-2019>Q 3

If a quadrilateral

A B C D

is circumscribed about a circle with centre

O

, prove that

A B + C D = B C + D A

EASY

10th West Bengal Board

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 4 - Theorems Regarding Tangent to a Circle>West Bengal Board-2020>Q 4

The length of the radii of two circles are

8 cm

and

3 cm

, the distance between the centres is

13 cm

. What is the length of the direct common tangent of the two circles?

MEDIUM

10th West Bengal Board

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 4 - Theorems Regarding Tangent to a Circle>West Bengal Board-2020>Q 5

Prove that if two tangents are drawn to a circle from a point outside it, then they make equal angles at the centre.

MEDIUM

10th West Bengal Board

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 4 - Theorems Regarding Tangent to a Circle>West Bengal Board-2018>Q 6

The radius of a circle with centre $O$ is $5 c m$ . $P$ is a point at a distance $13 c m$ from $O$ . $P Q$ and $P R$ are two tangents to this circle. Find the area of the quadrilateral $P Q O R$ .

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