Mensuration(Quantitative Aptitude ) Questions and answers for exam Preparation

Quantitative Aptitude

MENSURATION MCQs

Regular Polygons, Triangles, Circles

Total Questions : 254 | Page 7 of 26 pages

Question 61. The region inscribed by a chord and two radii having endpoints at the end of the chord will be a/an ___.

isosceles triangle
scalene triangle
square
parallelogram

Discuss Question

Answer: Option A. -> isosceles triangle
:
A
The shaded triangle in the figure is an isosceles triangle.
A triangle in which two sides are equal is called an isosceles triangle. Here, the two equal sides are the radii of the circle OA and OB.

The Region Inscribed By A Chord And Two Radii Having Endpoin...

Question 62. What will be the radius of a circle whose diameter is 36 cm?

11 cm
66 cm
16 cm
18 cm

Discuss Question

Answer: Option D. -> 18 cm
:
D
Given that
The length of the diameter of a circle = 36 cm
We know that, radius of a circle =

\frac{D i a m e t e r}{2}

On substituting the values we get:
Radius of circle =

\frac{36}{2} = 18 c m

Question 63. The figure is of a circle with its center at O. The line PQ is the:

Diameter of the circle
Radius of the circle
Chord of the circle
Tangent of the circle

Discuss Question

Answer: Option A. -> Diameter of the circle
:
A
The line PQ is the diameter of the circle. A diameter of a circle is a line segment that passes through centre of the circle and whose endpoints lie on the circle.

Question 64. In the adjoining figure, if AD, AE and BC are tangents to the circle at D, E and F respectively. Then

AD=AB+BC+CA
2AD=AB+BC+CA
3AD=AB+BC+CA
4AD=AB+BC+CA

Discuss Question

Answer: Option B. -> 2AD=AB+BC+CA
:
B

We know that

A D = A E

A D = A B + B E

Since

B E = B F

as tangents drawn from an external point to a circle are equal ,

A D = A B + B F

……(1)
Also

A D = A C + C D

A D = A C + C F

……(2) (

C D

and

C F

are tangents drawn from the external point C to the circle)
Adding equation (1) and (2),

A D + A D = A B + B F + C F + A C

2 A D = A B + B C + A C

Question 65. The length of a tangent from a point Q to a circle is 24 cm. The distance between Q and the center of the circle is 25 cm. The radius of the circle is _____ cm.

Discuss Question

Given that PQ = 24 cm and OQ = 25 cm.
To find OP, applyPythagoras theorem for

△

OPQ.

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

{O P}^{2} + 24^{2} = 25^{2}

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

\Rightarrow O P = 7 cm

Question 66. In the given figure, the circle touches the sides AB, BC, CD and DA of a quadrilateral ABCD at the points P, Q, R, S respectively. If AB = 11 cm, BC =x cm, CR = 4 cm and AS = 6 cm, the value of x is
___ cm.

Discuss Question

:
AP = AS = 6 cm
BP = BA – AP = 11 – 6 = 5
BQ = BP = 5
CQ = CR = 4
x = CQ + BQ
x = 4 + 5 = 9 cm

Question 67. Three circles touch each other externally. The distance between their centres is 5 cm, 6 cm and 7 cm. Find the radii of the circles.

2 cm, 3 cm, 4 cm
3 cm, 4 cm, 1 cm
1 cm, 2.5 cm, 3.5 cm
1 cm, 2 cm, 4 cm

Discuss Question

Answer: Option A. -> 2 cm, 3 cm, 4 cm
:
A
Consider the below figure wherein three circles touch each other externally.

Three Circles Touch Each Other Externally. The Distance Betw...

Since the distances between thecentres of these circles are5 cm, 6 cm and 7 cm respectively, we have the following set of equations with respect to the above diagram:
x+y = 5 …..(1)
y+z = 6 ......(2) (

\Rightarrow

y=6-z)... (2.1)
x+z = 7 …..(3)
Adding (1), (2) and (3), we have

2 (x + y + z) = 5 + 6 + 7 = 18

⟹ x + y + z = 9.... (4)

Using (1) in (4), we have

5 + z = 9 ⟹ z = 4

Now using,

(3) ⟹ x = 7 - z = 7 - 4 = 3

And

(2.1) ⟹ y = 6 - z = 6 - 4 = 2

Therefore, the radii of the circles are 3cm, 2cm and 4 cm.

Question 68. State true or false.
PQ is a tangent drawn from a point P to a circle with center O and QOR is a diameter of the circle such that ∠POR =

120^{\circ}

, then ∠OPQ is

30^{\circ}

True
False

Discuss Question

Answer: Option A. -> True
:
A

State True Or False.PQ Is A Tangent Drawn From A Point P To ...

Given that

∠ P O R = 120^{\circ}

We know that

∠

OQP is

90^{\circ}

Through external angle theorem

∠ P O R = ∠ O Q P + ∠ O P Q

\Rightarrow 120^{\circ} = 90^{\circ} + ∠ O P Q

∠ O P Q = 120^{\circ} - 90^{\circ}

∠ O P Q = 30^{\circ}

Question 69. Two concentric circles of radii a and b (a > b) are given. Find the length of the chord of the larger circle which touches the smaller circle.

√a2−b2
√a2+b2
2√a2−b2
2√a2+b2

Discuss Question

Answer: Option C. -> 2√a2−b2
:
C
Let O be the common centre of the two circles and AB be the chord of the larger circle which touches the smaller circle at C.
Join OA and OC.
Then OC

⊥

AB
LetOA = a and OC = b.

Two Concentric Circles Of Radii A And B (a > B) Are Given...

SinceOC

⊥

AB, OC bisects AB
[

∵

perpendicular from the centre to a chord bisects the chord].
In right

Δ

ACO, we have

O A^{2} = O C^{2} + A C^{2}

[by Pythagoras' theorem]

\Rightarrow A C = \sqrt{O A^{2} - O C^{2}} = \sqrt{a^{2} - b^{2}}

∴ A B = 2 A C = 2 \sqrt{a^{2} - b^{2}}

[

∵

C is the midpoint of AB]
i.e., Length of the chord

A B = 2 \sqrt{a^{2} - b^{2}}

Question 70. A circle is inscribed in a triangle with sides 3, 4 and 5 cm. The radius of the circle is
___ cm.

Discuss Question

:
Join the centre of the circle and the vertices of the triangle. Observe that the sides of the triangle become the tangents to the circle. Hence, the radii of the circle as shown in the question become the heights of the smaller triangles.