Mensuration(Quantitative Aptitude ) Questions and answers for exam Preparation

Quantitative Aptitude

MENSURATION MCQs

Regular Polygons, Triangles, Circles

Total Questions : 254 | Page 8 of 26 pages

Question 71. A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the major arc.

150∘
30∘
60∘
90∘

Discuss Question

Answer: Option B. -> 30∘
:
B

A Chord Of A Circle Is Equal To The Radius Of The Circle. Fi...

Given that AO = AB=OB.
Since all sides are equal,

△ A O B

is equilateral, and hence equiangular. Also, each angle of the triangle equals

60^{\circ} .

i.e.,

∠

AOB =

60^{\circ}

⟹ ∠ A C B = \frac{1}{2} A O B

(

∵

Angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.)

⟹ ∠ A C B = \frac{60^{\circ}}{2} = 30^{\circ}

Question 72. Two parallel lines touch a circle at points A and B respectively. The area of the circle is 25

π

{c m}^{2}

, then the distance between the lines is

5 cm
8 cm
10 cm
25 cm

Discuss Question

Answer: Option C. -> 10 cm
:
C
Given that the area of circle =

25 π c m^{2}

⟹ π r^{2} = 25 π

, where

r

is the radius of the circle.

i . e ., r^{2} = 25 ⟹ r = \pm 5

Since radius is a non-negative quantity, we have

r = 5 c m

Now, the distance between the twoparallel lines
= Diameter of the circle
=

2 \times

Radius of the circle = 10cm.

Two Parallel Lines Touch A Circle At Points A And B Respecti...

Question 73. Tangents PA and PB are drawn from an external point P to two concentric circles with centre O and radii 8 cm and 6 cm respectively, as shown in the figure. If AP = 6 cm then find the length of BP.

8 cm
16 cm
10 cm
6 cm

Discuss Question

Answer: Option A. -> 8 cm
:
A
We have
OA

⊥

AP and OB

⊥

BP [ The tangent at any point of a circle is perpendicular to the radius through the point of contact].
Join OP.

In right

Δ

OAP, we have
OA = 8 cm, AP = 6cm

∴ O P^{2} = O A^{2} + A P^{2}

[by Pythagorastheorem]

\Rightarrow O P = \sqrt{O A^{2} + A P^{2}} = \sqrt{8^{2} + 6^{2}} c m = \sqrt{100} c m = 10 c m

In right

Δ

OBP, we have
OB = 6cm, OP = 10cm

∴ O P^{2} = O B^{2} + B P^{2}

[by Pythagoras' theorem]

\Rightarrow B P = \sqrt{O P^{2} - O B^{2}} = \sqrt{10^{2} - 6^{2}} c m = \sqrt{64} c m

Thus, the length of BP

= \sqrt{64} c m

= 8cm.

Question 74. In the adjoining figure 'O' is the center of circle,

∠

CAO =

25^{\circ}

and

∠

CBO =

35^{\circ}

. What is the value of

∠

AOB?

In The Adjoining Figure 'O' Is The Center Of Circle, ∠CAO ...

55∘
110∘
120∘
Data insufficient

Discuss Question

Answer: Option C. -> 120∘
:
C

Δ A O C

,
OA=OC --------(radii of the same circle)

∴ Δ A O C

is an isosceles triangle

\to ∠ O A C = ∠ O C A = 25^{\circ}

----- (base angles of an isosceles triangle )
In

Δ B O C

,
OB=OC --------(radii of the same circle)

∴ Δ B O C

is an isosceles triangle

\to ∠ O B C = ∠ O C B = 35^{\circ}

-----(base angles of an isosceles triangle )

∠ A C B = 25^{\circ} + 35^{\circ} = 60^{\circ}

∠ A O B = 2 \times ∠ A C B

----(angle at the center is twice the angle at the circumference)
= 2

\times 60^{\circ}

120^{\circ}

Question 75. A line that touches a circle at only one point is called ________.

Discuss Question

:
A line that touchesa circle at only one point is called a tangent.

Question 76. A point P is 13 cm from the centre of the circle. The length of the tangent drawn from P to the circle is 12cm. Find the radius of the circle.

10 cm
5 cm
7 cm
12 cm

Discuss Question

Answer: Option B. -> 5 cm
:
B

A Point P Is 13 Cm From The Centre Of The Circle. The Length...

Since,
tangent to a circle is perpendicular to the radius through the point of contact
So,

∠ O T P = 90^{0}

So, in triangle OTP

(O P)^{2} = (O T)^{2} + (P T)^{2}

13^{2} = (O T)^{2} + 12^{2}

(O T)^{2} = 13^{2} - 12^{2}

O T^{2} = 25

O T = \sqrt{25}

O T = 5

So, radius of the circle is 5 cm

Question 77. A point P is 25 cm from the centre of a circle. The radius of the circle is 7 cm and length of the tangent drawn from P to the circle is x cm. The value of x is ___ cm.

Discuss Question

Given that OP = 25 cm and OQ = 7 cm.
To find the length of PQ, applyPythagoras theorem in

△

OPQ since

∠ O Q P = 90^{\circ}

{O Q}^{2} + {Q P}^{2} = {O P}^{2}

7^{2} + {Q P}^{2} = 25^{2}

{Q P}^{2} = 625 - 49 = 576

Q P = 24 c m

The length of the tangent is 24 cm.

Question 78. The maximum number of common tangents that can be drawn for two external touching circles is/are ___.

Discuss Question

:
Only three tangents can be drawn passing through two external touching circles.

The Maximum Number Of Common Tangents That Can Be Drawn For ...

Question 79. The line drawn through the centre of a circle to bisect a chord is __ to the chord.

Discuss Question

Any line passing through the centre of the circle will perpendicularly bisect the chord of thecircle.

Question 80. How many circles can be drawn through two points?

One
Two
Infinite
Four

Discuss Question

Answer: Option C. -> Infinite
:
C
An infinite number of circles can be drawn passing through two different points.
The line joining these two points can be the chord of an infinite number of circles having different radius.