Ncert – Class 10 – Circles– Hots Question

Question 1

Meena was studying Indian music and she happened to know about the Baul, a group of
mystic minstrels (musicians). Learnt about ektara, an instrument used by them. She started
studying the geometric structure of the instrument. If she represented the circular base and
sides by the following figure for this ektara by geometric diagram given below. Find a
relation between OP and radius of the circle.

Solution

From point P, two tangents are drawn.
It is given that, OT = a
And line OP bisects ∠RPT.
So,
∠TPO = ∠RPO = 45o
We know that, OT ⊥ PT
In right-angled triangle OTP,
Sin 45o = OT/OP
= 1/√2 = a/OP
Hence, OP = a√2

Question 2

Here is a picture of circular road from above. Considering it as a circle a diagram has been drawn
for reference on map (1cm = 100 m). Find the length BP.

Solution

PB= √127𝑐𝑚, So length of the line on land will be √127 𝑋 100 𝑚
= 11.27 X100m
=1127 m

Question 3

In the given figure, a circle touches all the four sides of a quadrilateral ABCD whose sides are AB = 6 cm, BC = 9 cm and CD = 8 cm. Find the length of side AD.

Solution

AD = 5 cm

Question 4

In the given figure, PA and PB are tangents to the circle from an external point P. CD is another tangent touching the circle at Question If PA = 12 cm, QC =QD = 3 cm, then find PC + PD.

Solution

PC + PD = 18 cm

Question 5

A circles touches the side QR of a △PQR at ‘M’ and side PQ and PR on producing at ‘S’ and ‘T’
respectively . If PS=8cm. Find the perimeter of △PQR.

Solution

.PQ+QS=8
PQ+QM=8
PS=8cm
PT=8
PR+RT=8
PR+RM=8
Perimeter of triangle PQR=16cm

Question 6

In the figure, PQRS is a quadrilateral such that S=90. A circle with centre O and radius r touches
the sides PQ, QR, RS and SP at A,B,C and D respectively . If QR=38cm, SR=28 cm and AQ=27cm, Find r.

Solution

QB+BR=38cm
BR=11cm
RC=11cm
SC+CR=28cm
SC=17
COD=90
OC=OD=r
rectangle OCSD is a square
OC=CS=17cm

Question 7

If a circle touches the side BC of a tringle ABC at P and extended sides AB and AC at Q and R
respectively . Prove that AQ=1/2 (BC+CA+AB)

Solution

.2AQ=(AB+BP)+(AC+CP)
=BC+CA+AB
AQ=1/2(BC+CA+CP)

Question 8

If a hexagon circumscribes a circle, show that sum of three alternate sides of the
hexagon is equal to the other three alternate sides.

Solution

AP + BP + CR+ DR+ ET + FT = AU + BQ + CQ + DS + ES + FU
AB + CD + EF = BC + DE+ AF

Question 9

Two wheels of a bicycle with centre O and O’ touches the road at two points, say P and Q. The distance between P and Q is 12 units. The distance of P from O’ is 13 units and distance of Q from O is 15 units. Find the radius of both the wheels.

Solution

4.Two wheels of a bicycle with centre O and O’ touches the road at two points, say P and Q. The distance between P and Q is 12 units. The distance of P from O is 13 units and distance of Q from O’ is 15 units. Find the radius of both the wheels.

Join OP and O’Q
In △O’PQ,
O’P2 = OP2 + PQ2
O’Q2 = 225 – 144
O’Q = 9 cm
Similarly, OP
= 5 cm

Question 10

If the angle between two tangents drawn from an external point P to a circle of radius a and centre O, is 60°, then find the length of OP.

Solution

Hence OP= 2 x radius of the circle