# prove that the line segment joining the point of contact of 2 parallel tangents to a circle is a diameter of the circle.

### Mohammed

Guys, does anyone know the answer?

get prove that the line segment joining the point of contact of 2 parallel tangents to a circle is a diameter of the circle. from screen.

## Prove that the line segment joining the point of contact of two parallel tangents to a circle is a diameter of the circle.

Prove that the line segment joining the point of contact of two parallel tangents to a circle is a diameter of the circle.

Byju's Answer Standard X Mathematics

Relations Using Incircle and Outcircle

Prove that th... Question

Prove that the line segment joining the point of contact of two parallel tangents to a circle is a diameter of the circle.

Open in App Solution

Given : CD and EF are two parallel tangents at the points A and B of a circle with center O.

To prove : AOB is a diameter of the circle

Construction : Join OA and OB

Draw OG | | CD

Proof : OG | | CD and AO cuts them .

⇒ 90 ∘ + GOA = 180 ∘

[ OA is perpendicular to CD ]

⇒ GOA = 90 ∘ ) Similarly, GOB = 90 ∘ ;

Therefore, GOA + GOB = (

90 ∘ + 90 ∘ ) = 180 ∘ )

=> AOB is a straight line

Hence, AOB is a diameter of the circle with center O.

Suggest Corrections 24

SIMILAR QUESTIONS

**Q.**

Prove that the line segment joining the points of contact of two parallel tangents of a circle, passes through its centre.

**Q.**Prove that the line joining the points of contact of two parallel tangents of a circle passes through its centre. [CBSE 2014]

**Q.**

Prove that the line segment joining the point of contact of two parallel tangles of a circle passes through its centre.

**Q.**

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segments joining the points of contact to the center

RELATED VIDEOS

MATHEMATICS Watch in App EXPLORE MORE

Relations Using Incircle and Outcircle

Standard X Mathematics

## Prove that the line segment joining the points of contact of two parallel tangents to a circle is a diameter of the circle.

Click here👆to get an answer to your question ✍️ Prove that the line segment joining the points of contact of two parallel tangents to a circle is a diameter of the circle.

Question

## Prove that the line segment joining the points of contact of two parallel tangents to a circle is a diameter of the circle.

Hard Open in App Solution Verified by Toppr

Consider the circle with center at O

PQ & RS are two parallel tangents to it touching at A and B respectively.

Join OA and OB

Now OA perpendicular to OQ (∴ radius is perpendicular to tangent)

And OB perpendicular to RS

∴OA∥OB

But OA and OB pass through O

∴AB is straight line through center

∴AB is a diameter

Was this answer helpful?

138 56

## Prove that the line segment joining the points of contact of two parallel tangents to a circle is a diameter of the circle.

Q.1 of chapter 10, 10. Circles - Maths Important Questions book. Prove that the line segment joining the points of contact of two parallel tangents to a circle is a diameter of the circle.

## Book: Maths Important Questions

### Chapter: 10. Circles

### Subject: Maths - Class 10th

### Q. No. 1 of Important Questions

Listen NCERT Audio Books to boost your productivity and retention power by 2X.

1

Given: and are the tangent to a circle such that || , intersecting at A and B respectively.

To prove: AB is a diameter of the circle.

Proof:

A tangent at any point of a circle is perpendicular to the radius through the point of contact.

∴ ∠ XAO = 90° and ∠ YBO = 90°

Since ∠ XAO + ∠ YBO = 180°

Angles on the same side of the transversal is 180°.

Hence the line AB passes through the centre and is the diameter of the circle.

## Chapter Exercises

### Important Questions

### Very Short Answer (1 Mark)

### Short Answer (2 Marks)

### Short Answer (3 Marks)

### Long Answer (4 Marks)

## More Exercise Questions

1

2

### Find the length of the tangent from a point which is at a distance of 5 cm from the centre of the circle of radius 3 cm.

3

### Find the locus of centres of circles which touch a given line at a given point.

4

### In two concentric circles, prove that all chords of the outer circle which touch the inner circle are of equal length.

5

### Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord.

6

### Find the locus of centres of circles which touch two intersecting lines.

Let A be one point of intersection of two intersecting circles with centres O and Q. The tangents at A to the two circles meet the circles again at B and C, respectively. Let the point P be located so that AOPQ is a parallelogram. Prove that P is the circumcentre of the triangle ABC in figure.7

(Hint: AQ ⊥ AB and AQ || OP. Then OP ⊥ AB and is also bisector of AB. Similarly, PQ is perpendicular bisector of AC.)

The radius of the incircle of a triangle is 4 cm and the segments into which one side is divided by the point of contact are 6 cm and8

8 cm. Determine the other two sides of the triangle.

(Hint: Equate the areas of the triangle found by using the formula √[s(s-a)(s-b)(s-c)] and also found by dividing it into three triangles.)

The radii of two concentric circles are 13 cm and 8 cm. AB is a diameter of the bigger circle. BD is a tangent to the smaller circle touching it at D. Find the length AD in figure. [Hint: Let line BD intersect the bigger circle at E. Join AE. AE = 2 × 8= 16 cm.9

DE = BD = √(169 - 64) = � and ∠AED= 90°.]

10

### Find the locus of the centre of a circle of constant radius (r) which touches a given circle of radius r1 (i) externally, (ii) internally, given r1> r.

In figure, two circles with centres O, O' touch externally at a point A. A line through A is drawn to intersect these circles at B and C. Prove that the tangents at B and C are parallel. [Hint: Prove that ∠ OBA = ∠ O' CA]11

In figure two circles intersect at two points A and B. From a point P on a circle, two line segments PAC and PBD are drawn intersecting the other circle at the points C and D. Prove that CD is parallel to the tangent at P.12

13

### Two rays ABP and ACQ are intersected by two parallel lines in B, C and P, Q respectively. Prove that the circumcircles of Δ ABC and Δ APQ touch each other at A. [Hint: Draw tangent XAY to the circumcircle of triangle APQ and show that ∠ YAP = ∠ PQA = ∠ BCA.

In figure, two circles touch internally at a point P. AB is a chord of the bigger circle touching the other circle at C. Prove that PC bisects the angle APB.[Hint: Draw a tangent at the point P. Joint CD, where D is the point of intersection of AP and the inner circle and prove that ∠ PBC = ∠ PCD.]14

15

### Two circles intersect each other at two points A and B. At A, tangents AP and AQ to the two circles are drawn which intersect other circles at the points P and Q respectively. Prove that AB is the bisector of angle PBQ.

16

Guys, does anyone know the answer?