CBSE Class 10-Mathematics: Questions and Answers Chapter –10 Circles Part 17

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Question 17:

Two tangents TP and TQ are drawn to a circle with center O from an external point T. Prove that .

A Circle with Centre O

A Circle with Centre O

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Answer:

Given: A circle with center O and an external point T and two tangents TP and TQ to the circle, where P and Q are the points of contact

To prove:

Proof: Let

In , we have

[Length of the tangents drawn from an external point to a circle are equal]

So, TPQ is an isosceles triangle.

In we have

But

[Angle between the tangent and radius of a circle is ]

Now

Question 18:

Prove that the parallelogram circumscribing a circle is a rhombus

ABCD be the Parallelogram

ABCD be the Parallelogram

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Answer:

Given: ABCD be the parallelogram circumscribing a circle with centre O such that the sides AB, BC, CD and DA touch a circle at P, Q, R and S, respectively.

To prove: is a rhombus.

Proof: …(i)

…(ii)

…(iii)

…(iv)

[Tangents drawn from an external point to a circle are equal]

Adding (i), (ii), (iii) and (iv), we get

Question 19:

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

tangents drawn at the ends of a chord

Tangents Drawn at the Ends of a Chord

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Answer:

Let NM be chord of circle with center C.

Let tangents at M.N meet at the point O.

Since OM is a tangent

is a tangent

Again in

Thus, tangents make equal angle with the chord.