Mensuration(Quantitative Aptitude ) Questions and answers for exam Preparation

Quantitative Aptitude

MENSURATION MCQs

Regular Polygons, Triangles, Circles

Total Questions : 254 | Page 4 of 26 pages

Question 31. Two tangents PA and PB are drawn from an external point P to the circle with centre O, such that

∠

APB =

120^{\circ}

what is the relation between OP and AP?

OP = 12 AP
AP = 23 OP
OP = 2 AP
OP = AP

Discuss Question

Answer: Option C. -> OP = 2 AP
:
C

Two Tangents PA And PB Are Drawn From An External Point P t...

Given that

∠ A P B = 120^{\circ}

Also, we know that if two tangents are drawn from an external point to a circle, then the line joining the external point and the centre of the circle bisects the angle between the tangents.

⟹ ∠ A P O = ∠ O P B = 60^{\circ}

Thus, cos

∠ O P A = c o s 60^{\circ} = \frac{A P}{O P}

⟹ \frac{1}{2}

\frac{A P}{O P}

Thus,

O P = 2 A P

Question 32. If TP and TQ are two tangents to a circle with center O such that

∠ P O Q = 110^{\circ}

, then,

∠

PTQ is equal to:

If TP And TQ Are Two Tangents To A Circle With Center O Such...

60∘
70∘
80∘
90∘

Discuss Question

Answer: Option B. -> 70∘
:
B

Given that

∠ P O Q = 110^{\circ}

Note that

∠ O Q T = ∠ O P T = 90^{\circ}

(

∵

A tangent at any point of a circle is perpendicular to the radius at the point of contact)
Also,

∠ T Q O + ∠ Q O P + ∠ O P T + ∠ P T Q = 360^{\circ}

(

∵

Sum of interior angles of a quadrilateral is 360 degrees)

⟹ ∠ P T Q = 360^{\circ} - 90^{\circ} - 90^{\circ} - 110^{\circ}

⟹ ∠ P T Q = 70^{\circ}

∴ ∠ P T Q = 70^{\circ}

Question 33. Through any given set of three distinct points A, B, C it is possible to draw at most ___circle(s).

Discuss Question

:
At most one circle can be drawn through a given set of three distinct points. These threepoints will then be referred to as 'concyclic points' (Lying on the same circle). .

Question 34. In the given figure, AB is the diameter of the circle. Find the value of

x

30∘
45∘
60∘
90∘

Discuss Question

Answer: Option D. -> 90∘
:
D
An angle formed by the diameter of a circle at its circumference equals

90^{\circ} .

Hence, the value of

x

90^{\circ}

Question 35. The length of the complete circle is called __ of the circle.

Discuss Question

:
The boundary of the circle is called its circumference and the value of circumference is

2 π r (where r is the radius of thecircle)

. If we cut a circle and form a line from it, then the length of the line will be the same as the circumference of the circle.

Question 36. Only one circle can be drawn through three non-collinear points.

True
False
Segment
Arc

Discuss Question

Answer: Option A. -> True
:
A
With three non-collinear points, onlyone circle can be drawn as shown below. Note the points should be non-collinear otherwise no circle can be drawn.

Only one Circle can Be Drawn through Three Non-collinear ...

Question 37. ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If

∠ D B C = 80^{\circ}

and

∠ B A C = 40^{\circ},

then find

∠ B C D

ABCD Is A Cyclic Quadrilateral Whose Diagonals Intersect At ...

60∘
80∘
50∘
40∘

Discuss Question

Answer: Option A. -> 60∘
:
A
Given that

∠ B A C = 40^{\circ} and ∠ D B C = 80^{\circ} .

Since the angles formed by the same segment are equal,

∠ B D C = ∠ B A C = 40^{\circ} .

In Δ B D C,

∠ B D C + ∠ D B C + ∠ B C D = 180^{\circ} . [Angle sum property]

i.e.,

40^{\circ} + 80^{\circ} + ∠ B C D = 180^{\circ}

⟹ ∠ B C D = 180^{\circ} - 120^{\circ} = 60^{\circ}

Question 38. A tangent is drawn at a point P on a circle. A line through the centre O of a circle of radius 7 cm cuts the tangent at Q such that PQ = 24 cm. Find OQ.

10 cm
15 cm
20 cm
25 cm

Discuss Question

Answer: Option D. -> 25 cm
:
D
Since tangent at a point on the circle is perpendicular to the radius throughthatpoint of contact.

A Tangent Is Drawn At A Point P On A Circle. A Line Through ...

∴ O P ⊥ O Q

Inrightangledtriangle OPQ,

{O Q}^{2} = {O P}^{2} + {P Q}^{2} (ByPythagorastheorem)

{O Q}^{2} = 7^{2} + 24^{2}

{O Q}^{2} = 49 + 576

{O Q}^{2} = 625

O Q = \sqrt{625} = 25 c m

Question 39. An equilateral triangle ABC is inscribed in a circle with centre O. Then,

∠ B O C

is equal to ___.

An Equilateral Triangle ABC Is Inscribed In A Circle With Ce...

30∘
60∘
90∘
120∘

Discuss Question

Answer: Option D. -> 120∘
:
D
Given that

Δ A B C

is equilateral.

⟹ ∠ B A C = 60^{\circ}

Since the angle subtended by a chord at the centre of a circle is twice the angle subtended by the same chord at any other point on the remaining part of the circle, we have

∠ B O C = 2 ∠ B A C = 2 \times 60^{\circ} = 120^{\circ} .

Question 40. A circle with center O and radius 5 cm has two chords AB and AC, such that AB = AC = 6 cm. If by joining B and C the line segment passes from the center of the circle O, then what is the length of BC?

5 cm
9 cm
10 cm
11.9 cm

Discuss Question

Answer: Option C. -> 10 cm
:
C
As the line segment BC passes through the center, it means that it is a diameter of the circle.
Given that radius of the circle = 5 cm