MCQ Questions on Geometry for Competitive Exams

61

Let D and E be two points on the side BC of ΔABC such that AD = AE and ∠BAD = ∠EAC. If AB = (3x + 1) cm, BD = 9 cm, AC = 34 cm and EC = (y + 1) cm, then the value of (x + y) is:

A) 19 B) 16 C) 17 D) 20

Discuss

Answer & Solution

Answer: Option A

Solution:

ΔABC Isosceles Δ
3x + 1 = 34
x = 11
From Pythagoras Triplet
AP = 30 cm
PC = BP = 16 cm
DP = 7 cm
PC = 7 + y + 1
16 = 8 + y
y = 8
x + y = 11 + 8 = 19

62

In the given figure, PQRS is a square inscribed in a circle of radius 4 cm. PQ is produced till point Y. From Y a tangent is drawn to the circle at point R. What is the length (in cm) of SY?
Geometry mcq question image

A) $$4\sqrt {10} $$ B) $$2\sqrt {10} $$ C) $$6\sqrt {10} $$ D) $$3\sqrt 5 $$

Discuss

Answer & Solution

Answer: Option A

Solution:

∠PRQ = 45°
∠ORY = 90°
∠QRY = 45°
RQ = QY
Diagonal = 8 = Diameter
Side $$ = \frac{8}{{\sqrt 2 }} = 4\sqrt 2 $$
In ΔSPY,
SY² = SP² + PY²
= 128 + 32
= 160
SY = $$4\sqrt {10} $$

63

ABC is an isosceles triangle where AB = AC which is circumscribed about a circle. If P is the point where the circle touches the side BC, then which of the following is true?

A) BP = PC B) BP > PC C) BP < PC D) BP = $$\frac{1}{2}$$PC

Discuss

Answer & Solution

Answer: Option A

Solution:

Here given that = AB = AC
AQ + BQ = AR + RC
We know that
BQ = PB & PC = RC
AQ + PB = AR + PC
Also AQ = AR
AR + PB = AR + PC
PB = PC

64

AB is a diameter of a circle with centre O, CB is a tangent to the circle at B. AC intersects the circle at G. If the radius of the circle is 6 cm and AG = 8 cm, then the length of BC is:

A) 2√5 cm B) 6√6 cm C) 2√6 cm D) 6√5 cm

Discuss

Answer & Solution

Answer: Option D

Solution:

$$\eqalign{ & \Delta AGB \sim \Delta BGC \cr & \frac{{AB}}{{BC}} = \frac{{BG}}{{GC}} = \frac{{AG}}{{BG}} \cr & \frac{{12}}{{BC}} = \frac{{4\sqrt 5 }}{{GC}} = \frac{8}{{4\sqrt 5 }} \cr & GC = 10 \cr & \frac{{12}}{{BC}} = \frac{{4\sqrt 5 }}{{10}} \cr & BC = 6\sqrt 5 \cr} $$

65

A circle is inscribed in a triangle ABC. It touches side AB, BC and AC at points R, P and Q, respectively. If AQ = 2.6 cm, PC = 2.7 cm and BR = 3 cm, then the perimeter (in cm) of the triangle ΔABC is:

A) 28 B) 30 C) 16.6 D) 33.2

Discuss

Answer & Solution

Answer: Option C

Solution:

Because tangent drawn from a single point are equal.
So perimeter of ABC = (2.6 + 2.7 + 3) × 2 = 16.6

66

PQR is a triangle. S and T are the midpoints of the sides PQ and PR respectively. which of the following is TRUE?
I. Triangle PST is similar to triangle PQR.
II. ST = $$\frac{1}{2}$$ (QR)
III. ST is parallel to QR.

A) Only I and II B) Only II and III C) Only I and III D) All I, II and III

Discuss

Answer & Solution

Answer: Option D

Solution:

Triangle PST is similar to triangle PQR
Geometry mcq question image

[By AAA]
By mid point theorem.
ST = $$\frac{1}{2}$$(QR)
ST is parallel to QR.
All I, II and III are true.

67

B₁ is a point on the side AC of ΔABC and B₁B is joined. A line is drawn through A parallel to B₁B meeting BC at A₁ and another line is drawn through C parallel to B₁B meeting AB produced at C₁. Then

A) $$\frac{1}{{{\text{C}}{{\text{C}}_1}}} - \frac{1}{{{\text{A}}{{\text{A}}_1}}} = \frac{1}{{{\text{B}}{{\text{B}}_1}}}$$ B) $$\frac{1}{{{\text{C}}{{\text{C}}_1}}} + \frac{1}{{{\text{A}}{{\text{A}}_1}}} = \frac{1}{{{\text{B}}{{\text{B}}_1}}}$$ C) $$\frac{1}{{{\text{B}}{{\text{B}}_1}}} - \frac{1}{{{\text{A}}{{\text{A}}_1}}} = \frac{1}{{{\text{C}}{{\text{C}}_1}}}$$ D) $$\frac{1}{{{\text{B}}{{\text{B}}_1}}} + \frac{1}{{{\text{A}}{{\text{A}}_1}}} = \frac{1}{{{\text{C}}{{\text{C}}_1}}}$$

Discuss

Answer & Solution

Answer: Option B

No explanation is given for this question. Let's Discuss on Board

68

In a circle with centre O, ABCD is a cyclic quadrilateral and AC is the diameter. Chords AB and DC are produced to meet at E. If ∠CAE = 34° and ∠E = 30°, then ∠CBD is equal to:

A) 24° B) 36° C) 26° D) 34°

Discuss

Answer & Solution

Answer: Option C

Solution:

In, ΔBCE
∠B + ∠C + ∠E = 180°
90° + ∠C + 30° = 180°
∠C = 60°
In, ΔABC
∠A + ∠B + ∠C = 180°
34° + 90° + ∠C = 180°
∠C = 56°
∠ACD = 64° = ∠ABD
∠CBD = 90° - 64° = 26°

69

G is the centroid of the triangle ABC. where AB, BC and CA are 7 cm, 24 cm and 25 cm respectively. than BG is:

A) $$6\frac{1}{3}{\text{ cm}}$$ B) $$8\frac{1}{3}{\text{ cm}}$$ C) $$5\frac{1}{2}{\text{ cm}}$$ D) $$4\frac{1}{6}{\text{ cm}}$$

Discuss

Answer & Solution

Answer: Option B

Solution:

$$\eqalign{ & BD = \frac{{25}}{2} \cr & BG = BD \times \frac{2}{3} = \frac{{25}}{2} \times \frac{2}{3} = 8\frac{1}{3}{\text{ cm}} \cr} $$

70

Two circles touch externally. The sum of their areas is 130π sq cm and the distance between their centres is 14 cm. The radius of the smaller circle is:

A) 2 cm B) 3 cm C) 4 cm D) 5 cm

Discuss

Answer & Solution

Answer: Option B

Solution:

Let smallest circle radius = R
Then biggest circle radius = (14 - R)
Geometry mcq question image

According to the question,
⇒ π(14 - R)²2 + πR² = 130π
⇒ (14 - R)² + R² = 130
⇒ 196 + R² - 28R + R² = 130
⇒ R = 3 cm
⇒ Radius of smallest circle
R = 3 cm