NCERT Class 10 Chapter 11 Circles CBSE Board Sample Problems Long Answer

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Question

In given figure, AB is a chord of a circle, with centre O, such that and radius of circle is 10 cm. Tangents at A and B intersect each other at P. Find the length of PA

Circle

Circle

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Solution

Let

As OP is perpendicular bisector of AB. Then

In

( Pythagoras theorem]

In

(Pythagoras theorem)

In

Now,

From

, 16(X)

Question

Prove that the lengths of tangents drawn from an external point to a circle are Equal

Solution

Circle are Equal

Circle Are Equal

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Given: A circle is a point outside the circle and PA and PB arc tangents to a circle.

To Prove:

Construction: Draw OA, OB and OP

Proof: Consider triangles OAP and 01W

(i)

(Radius is perpendicular to the tangent at the point of contact)

OP is common ...(iii)

(From (i), (ii) and (iii))

Hence. (CPCT)

Question

In given figure, there are two concentric circles of radii 6 cm and 4 cm with centre O. If AP is a tangent to the larger circle and BP to the smaller circle and length of AP is 8cm, find the length of BP

Two concentric circles

Two Concentric Circles

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Solution

(Given radius)

(Given radius)

In (Pythagoras theorem)

Ln (Pythagoras theorem)

Question

In given figure, from a point P, two tangents PT and PS are drawn to a circle with centre O such that , Prove that

Two tangents PT and PS

Two Tangents PT and PS

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Solution

Let ( Tangent drawn from external point to circle arc equal)

In

( Each equal to 90°, since tangent perpendicular r radius)

(Equal radii)

[common]

(By SAS congruence rule]

[By CPCT]

Hence proved. X

Question

A quadrilateral is drawn to circumscribe a circle. Prove that the sums of opposite sides are equal.

Solution

A quadrilateral ABCD which circumscribe a circle, let it touches the circle at P. Q.

R and S as shown in fig.

To prove:

Proof: we know that the lengths of the tangents drawn from a point outside the circle to the circle are equal

Consider

Circumscribe a circle

Circumscribe a Circle

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Question

In fig, a circle is inscribed in a triangle PQR with and. find the length of the QM, RN and PL.

A circle

A Circle

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Solution

Let

Now,

Therefore

Hence,

Question

In the given fig, the sides AB, BC And AC of AABC touch a circle with center O and radius r at P, Q and R respectively. Prove that:

1)

2) Area

A circle

A Circle

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Solution

We have,

Area

Question

In figure, a Triangle ABC is drawn to circumscribe a circle of radius 3 cm, such that segments BD and DC are of length 6 cm and 9 cm. If area of the triangle ABC is, then find the length of the side AB and AC.

Circumscribe a circle of radius

Circumscribe a Circle of Radius

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Solution

Let

(tangents drawn from an external points are equal)

Also,

In Triangle

Area

On solving

Question

A triangle ABC is drawn to circumscribe a circle of radius 4cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6cm respectively. Find the sides AB and AC.

Solution

Circumscribe a circle of radius

Circumscribe a Circle of Radius

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In,

Length of two tangents drawn from the same point to the circle are equal,

We observed that,

Now semi perimeter of circles,

Area of

Also, Area of

Equating equation (I) and (li) we get,

Squaring both sides,

Sides are

Question

In the figure. XY and X’Y are two parallel tangents to a circle with Centre O and another tangent AB with point of contact C intersecting XV at A and X’Y’ at B. Prove that .

Two parallel tangents

Two Parallel Tangents

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Solution

Two parallel tangents

Two Parallel Tangents

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Join OC

In and

(radii of same circle)

(length of two tangents)

Common)

Therefore, (By SSS congruency criterion)

Hence. (CPCT)

Similarly

Now,

(Angle sum property)

Question

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

Prove that

Solution

Two tangents TP and TQ

Two Tangents TP and TQ

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To prove

TP and TO are tangents to the

Circle with centre O.

In quad OQTP

[Using angle sum property in ]

. Hence proved.

Question

ABC is a triangle. A circle touches sides AB and AC produced and side BC at Q, R and P respectively. Show that the perimeter of triangle ABC.

Triangle

Triangle

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Solution

Given: A circle touching the side BC of at P and AB, AC produced at Q and R respectively.

To Prove: (Perimeter of)

Proof: Lengths of tangents drawn from an external point to a circle are equal.

Perimeter o

(Perimeter of ) I

AQ is the half of the perimeter of .