NCERT Class 10 Chapter 11 Circles CBSE Board Sample Problems Long Answer (For CBSE, ICSE, IAS, NET, NRA 2022)
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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
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
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
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
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
Question
In fig, a circle is inscribed in a triangle PQR with and. find the length of the QM, RN and PL.
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
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.
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
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 .
Solution
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
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.
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 .