# PRACTICE QUESTIONS ON CIRCLES AND CYCLIC QUADRILATERALS

## About "Practice Questions on Circles and Cyclic Quadrilateral"

Practice Questions on Circles and Cyclic Quadrilateral :

Here we are going to see some example problems on circles and cyclic quadrilaterals.

## Practice Questions on Circles and Cyclic Quadrilateral - Practice questions

Question 1 :

In the given figure, AB and CD are the parallel chords of a circle with centre O. Such that AB = 8cm and CD = 6cm. If OM ⊥ AB and OL ⊥ CD distance between LM is 7cm. Find the radius of the circle.

Solution :

Let us join the points OC, OD, OA and OB.

OA  =  OB  =  OC  and OD are equal in length and it is radius.

Let OL  =  x

In triangle OLD,

CL  =  CD  =  3 cm

OL2 + LD2  =  OD2

x2 + 32  =  OD2

OD2  =  x2 + 9  ------------(1)

In triangle OAB,

OM  =  7 - x

AM  =  BM  =  4 cm

OM2 + MB2  =  OB2

(7 - x)2 + 42  =  OB2

OB2  =  (7-x)2 + 16  ------------(2)

(1)  =  (2)

(7-x)2 + 16  =  x2 + 9

49 - 14x + x2 + 16 - x2 - 9  =  0

-14x + 56  =  0

x  =  56/14  =  4

By applying the value of x in (1), we get

OD2  =  42 + 9

OD2  =  16 + 9  =  25

OD  =  5 cm

Question 2 :

The arch of a bridge has dimensions as shown, where the arch measure 2 m at its highest point and its width is 6 m. What is the radius of the circle that contains the arch?

Solution :

DB  =  2 m and DB  =  3 m

CE  =  CB  =  AC  (radius)

Let DC  =  x

In triangle CDB,

CB2  =  CD2 + DB2

CB2  =  x2 + 3 ---(1)

CE  =  x + 2

By applying the value of CE  =  x + 2 (CB) in (1), we get

(x + 2)2  =  x2 + 9

x2 + 4x + 4 - x2 - 9  =  0

4x - 5  =  0

x  =  5/4

x  =  1.25

CE  =  1.25 + 2  =  3.25 m

Question 3 :

In figure, <ABC =120 , where A,B and C are points on the circle with centre O. Find <OAC ?

Solution :

Question 4 :

A school wants to conduct tree plantation programme. For this a teacher allotted a circle of radius 6m ground to nineth standard students for planting sapplings. Four students plant trees at the points A,B,C and D as shown in figure. Here AB = 8m, CD = 10m and AB is perpendicular to CD. If another student places a flower pot at the point P, the intersection of AB and CD, then find the distance from the centre to P.

Solution :

AM  =  (1/2) AB

AM  =  (1/2)8  =  4

CN  =  (1/2) 10  =  5

OA  =  6

OA2  =  OM2 + MO2

62  =  OM2 + 42

36 - 16  =  OM2

OM  =  20  =  2√5

OD2  =  ON2 + ND2

62  =  ON2 + 52

36 - 25  =  ON2

ON  =  √11

OMPN is a rectangle and OP is a diagonal.

OP2  =  ON2 + PN2

OP2  =  √112 + (20)2

OP2  =  11 + 20

OP  =  √31  =  5.56 (approximately)

Question 5 :

In the given figure, <POQ = 100 and <PQR = 30, then find <RPO .

Solution :

After having gone through the stuff given above, we hope that the students would have understood, "Practice Questions on Circles and Cyclic Quadrilateral"

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