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

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## 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.

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

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## 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

138 56

स्रोत : www.toppr.com

## 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

### Q. No. 1 of Important Questions

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

1

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

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.

## More Exercise Questions

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### 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.

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(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 and

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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.

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

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### 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]

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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.

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### 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.]

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### The diagonals of a parallelogram ABCD intersect in a point E. Show that the circumcircles of Δ ADE and Δ BCE touch each other at E.

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Mohammed 3 day ago

Guys, does anyone know the answer?