Circles Test

Result

Circles Test - 23

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Weekly Quiz Competition

Question 1
1 / -0

If four sides of a quadrilateral $$ABCD$$ are tangential to a circle, then

A
$$AC+AD=BD+CD$$

B
$$AB+CD=BC+AD$$

C
$$AB+CD=AC+BC$$

D
$$AC+AD=BC+DB$$

Solution

A circle has been inscribed in a quadrilateral $$ABCD$$.
To find out- which of the options is true.
Solution :
Let the circle touch the side $$AB$$ at $$P$$, $$BC$$ at $$Q$$, $$CD$$ at $$R$$ and $$AD$$ at $$S$$.
$$\\ \therefore$$ $$ AP=AS, BP=BQ, CR=CQ$$ and $$ DR=DS $$, since the lengths of the tangents, drawn from a point to a circle, are equal.
$$ \therefore AP+BP=AB=AS+BQ$$ .....(i)
$$CR+DR=CD=CQ+DS$$ ......(ii).
Adding (i) and (ii), we get
$$ AB+CD=AS+BQ+CQ+DS=(AS+DS)+(BQ+CQ)=AD+BC$$
Ans : Option B
Question 2
1 / -0

If two circles touch externally, then number of common tangents is

A
$$2$$

B
$$3$$

C
$$1$$

D
$$0$$

Solution

$$\Rightarrow$$ In the diagram, we have two circles with center $$A$$ and center $$B$$ touching externally.

$$\Rightarrow$$ Tangents $$m,n$$ and $$l$$ are direct common tangent touches circle..

$$\therefore$$ If two circles touch externally, then the number of common tangents is $$3$$
Question 3
1 / -0

In Fig. 1, triangle ABC is circumscribing a circle. Then the length of BC is :

A
7cm

B
8cm

C
9cm

D
10cm

Solution

Since the tangents from an external point are equal
So
$$BN = BL$$
$$BL = 4 cm$$
$$CM = CL$$
$$CL = 6 cm$$
So, $$BC = CL + BL = 6 + 4$$
= $$ 10 cm$$
legth of BC is 10cm.
Question 4
1 / -0

In the given figure, quadrilateral $$ABCD$$ is circumscribed. The circle touches the sides $$AB,BC,CD,DA$$ at $$P,Q,R,S$$ respectively. If $$AP=9cm,BP=7cm,CQ=5cm, DR=6cm$$, find the perimeter of quad. $$ABCD$$

A
$$54cm$$

B
$$44cm$$

C
$$24cm$$

D
$$64cm$$

Solution

$$ABCD$$ is a quadrilateral and let $$O$$ be the center of the circle inscribed in the quadrilateral.
$$AB$$, $$BC$$, $$CD$$ and $$AD$$ are the tangent to the circle at $$P$$, $$Q$$, $$R$$ and $$S$$, $$AP=9$$, $$PB=7$$, $$QC=5$$ and $$DR=6$$
We know that the lengths of the two tangents from a point to circle are equal.
Thus, $$AP=AS=9$$ cm
$$BP=BQ=7$$ cm
$$CQ=CR=5$$ cm
$$DR=DS=6$$ cm
Perimeter of quadrilateral $$ABCD=AB+BC+CD+AD$$
$$=AP+BP+BQ+CQ+CR+DR+DS+AS$$
$$=9+7+7+5+5+6+6+9$$
$$=54$$ cm
Question 5
1 / -0

The number of common tangents that can be drawn to two non-intersecting circles is

A
$$4$$

B
$$3$$

C
$$2$$

D
$$1$$

Solution

There can be four tangents drawn to two non-intersecting circles from a point which lies outside of both circles.
Question 6
1 / -0

There is ___ tangent to a circle passing through a point lying inside the circle.

A
no

B
one

C
two

D
three

Solution

Tangent line to a circle is a line that touches the circle at exactly one point, never entering inside the circle. So, there is no tangent to a circle passing through a point lying inside the circle.
Option A is correct.
Question 7
1 / -0

How many common tangents can be drawn?

A
$$0$$

B
$$1$$

C
$$2$$

D
$$3$$

Solution

Only two direct common tangents can be drawn to the two touching circles.
So, option C is correct.
Question 8
1 / -0

If $$P$$ is a point, then how many tangents to a circle can be drawn from the point $$P$$, if it lies On the circle.

A
$$0$$

B
$$1$$

C
$$2$$

D
$$3$$

Solution

There is only one tangent to circle passing through a point lying on the circle.
Question 9
1 / -0

A tangent is a special case of ________ which touches circle at only one point.

A
Point of contact

B
Line

C
Secant

D
None

Solution

A tangent is special case of secant which touches circle at only one point.
Question 10
1 / -0

How many tangents can be drawn from a point outside the circle?

A
1

B
2

C
3

D
0

Solution

Two tangents can be drawn to a circle from a point outside the circle.
No tangent line can be drawn through a point within a circle, since any such line must be a secant line. However, two tangent lines can be drawn to a circle from a point P outside of the circle.