Maths MCQ Ch-6 Class 10 | Triangle

Mathematics

Multiple Choice Questions (MCQ)

Class 10 | Chapter 6 | Triangles

MCQ | CHAPTER 6 | TRIANGLES

Q 1) Which of the following triangles have the same side lengths?
(a) Scalene
(b) Isosceles
(c) Equilateral
(d) None of these 

Ans: c

Q 2) Two congruent figures are similar when two similar figures are congruent.
a) True
b) False
Ans: a
Explanation: Two geometric figures are said to be similar if they are of same shape but different sizes and congruent if they have same shape and size.

Q 3) The figure shown is congruent.

    

a) True
b) False
Ans: a
Explanation The stars in the given figure are congruent because they are same shape and same size. Congruent figures have same shape and size

Q 4) Two geometric figures which have same shape and size are known as congruent.
a) False
b) True 

Ans: b

Q 5) If ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R then, ∆ABC & ∆PQR are similar according to which test?
a) AAA test
b) AA test
c) SAS test
d) SSS test 

Ans: a

Q 6) The figure shown below is similar.

     

a) True
b) False
Ans: a
Explanation The two figures shown are similar because they have same shape but are different in sizes. The second pentagon is smaller in size as compared to a pentagon, but the basic structure of both the figures is same i.e. both are pentagon.

Q 7) In the given figure DE || BC, if AD = 5.9cm, DB = 4cm and AE = 7cm then, what will be the value of AC?

a) 2.3 cm        b) 5.1 cm            c) 11.74 cm            d) 10.9 cm 

Ans: c

Q 8) D and E are the midpoints of side AB and AC of a triangle ABC, respectively and BC = 6 cm. If DE || BC, then the length (in cm) of DE is:
(a) 2.5
(b) 3
(c) 5
(d) 6 

Ans: b

Q 9) A man goes 10 m west and 24 m north. Find his distance from the starting point.
a) 34 m
b) 28 m
c) 30 m
d) 26 m 

Ans: d

Q 10) What is the value of x id DE || BC ?

a) 0            b) 1            c) 2            d) 3 

Ans: b

Q 11) The diagonals of a rhombus are 16 cm and 12 cm, in length. The side of rhombus in length is:
(a) 20 cm
(b) 8 cm
(c) 10 cm
(d) 9 cm
Ans: c
Solution Hint
Since diagonals of a rhombus bisects each other at right angle
By Pythagoras theorem,
(16/2)² + (12/2)² = side²

8² + 6² = side²

64 + 36 = side² 

 Side = 10 cm

Q 12) The perimeters of two similar triangles ABC, PQR is 64 cm and 24 cm respectively. If PQ is 12 cm what will be the length of AB?
a) 30 cm
b) 32 cm
c) 12 cm
d) 16 cm
Ans: b
Solution Hint perimeters of similar triangles is the same as the ratio of their corresponding sides.

Q 13) If ∠D = ∠L, ∠E = ∠M then, ∆DEF & ∆LMN are similar according to which test?
a) AAA test
b) AA test
c) SAS test
d) SSS test 

Ans: b

Q 14) In the given figure, LM || PQ, what will be the relation between x, a, b and c?

a) a = c / b
b) ab = cx
c) bx = ac

d) cb = ax 

Ans: c

Q 15) Corresponding sides of two similar triangles are in the ratio of 2:3. If the area of small triangle is 48 sq.cm, then the area of large triangle is:
(a) 230 sq.cm.
(b) 106 sq.cm
(c) 107 sq.cm.
(d) 108 sq.cm 

Ans: d

Q 16) If DE || BC, AD = 4cm, BD = 7.5cm, AE = 6.4 cm & DE = 5cm then what will be the lengths of BC?

a) 11.23 cm
b) 15.24 cm
c) 14.375 cm

d) 14.275 cm 

Ans: c

Q 17) In ∆ ABC, DE || BC, AD = 4 cm, BD = 5 CM, DE = 8 CM Then BC =
a) 12 cm
b) 16 cm
c) 18 cm
d) 20 cm 

Ans: c

Q 18) Area of an equilateral triangle with side length a is equal to:
(a) √3/2 a
(b) √3/2 a²
(c) √3/4 a²
(d) √3/4 a

Ans: c

Q 19) If the areas of two similar triangles are in the ratio 361 : 529. What would be the ratio of the corresponding sides?
a) 19 : 23
b) 23 : 19
c) 361 : 529
d) 15 : 23 

Ans: a

Q 20) If perimeter of a triangle is 100 cm and the length of two sides are 30 cm and 40 cm, the length of third side will be:
(a) 30 cm
(b) 40 cm
(c) 50 cm
(d) 60 cm 

Ans: a

Q 21) In two similar triangles ∆ABC and ∆DEF, AB = 15cm, DE = 5cm. If AL and DM are the altitudes of the triangles ABC, DEF respectively, then what will be the ratio of their altitudes?

 

a) 3 : 1
b) 1 : 3
c) 1 : 2
d) 2 : 1 

Ans: a

Q 22) The perimeter of two similar triangles are 50 cm and 30 cm respectively. If one side of first triangle is 9 cm, then find the corresponding side of the other triangle.
a) 4.8 cm
b) 5.6 cm
c) 3.8 cm
d) 5.4 cm 

Ans: d

Q 23) In ∆ ABC, AB = 6√3 cm, AC = 12 cm and BC = 6 cm. The angle B is
(a) 120°
(b) 60°
(c) 90°
(d) 45° 

Ans: c

Q 24) If triangles ABC and DEF are similar and AB = 4 cm, DE = 6 cm, EF = 9 cm and FD = 12 cm, then find the perimeter of triangle ABC
(a) 22 cm
(b) 20 cm
(c) 21 cm
(d) 18 cm
Ans: d
Solution Hint: ABC ~ DEF
AB = 4 cm, DE = 6 cm, EF = 9 cm and FD = 12 cm
AB/DE = BC/EF = AC/DF
4/6 = BC/9 = AC/12
⇒ BC = (4 x 9)/6 = 6 cm
⇒ AC = (12 x 4)/6 = 8 cm
Perimeter = AB + BC + AC = 4 + 6 + 8 = 18 cm

Q 25) If the area of two similar triangles are in 36 : 49 then ratio of their corresponding sides be
a) 6 : 7
b) 7 : 6
c) 36 : 49
d) 49 : 36 

Ans: a

Q 26) The height of an equilateral triangle of side 5 cm is:
(a) 4.33 cm
(b) 3.9 cm
(c) 5 cm
(d) 4 cm 

Ans: a

Q 27) The area of two similar triangles are 36 cm2 and 25 cm2 respectively. If perimeter of first triangle is 9 cm then find the corresponding perimeter of the second triangle.
a) 8 cm
b) 8.5 cm
c) 7 cm
d) 7.5 cm 

Ans: d

Q 28) If D, E, F are the mid points of the sides AB, BC, and AC in triangle ABC then find Ar (∆DEF) : Ar (∆ABC)
a) 4 : 1
b) 1 : 2
c) 2 : 1
d) 1 : 4 

Ans: d

Q 29) If ABC and DEF are two triangles and AB / DE = BC / FD, then the two triangles are similar if
(a) ∠A = ∠F
(b) ∠B = ∠D
(c) ∠A = ∠D
(d) ∠B =∠E 

Ans: b

Q 30) What will be the length of the square inscribed in a circle of radius 5 cm?
a) 2.34 cm
b) 3.45 cm
c) 5√2 cm
d) 2.45 cm
Ans: c
Solution Hint

Radius = 5 cm, Diameter = 10 cm ⇒ AC = 10 cm
Let side of square = x ⇒ AB = BC = x
By Pythagoras theorem

AB² + BC² = AC²

x² + x² = 10² 

2x² = 100 ⇒ x² = 50 ⇒ x = 5√2

Q 31) What will be the value of BC if the area of two similar triangles ∆ABC and ∆PQR is 64cm² and 81cm² respectively and the length of QR is 15cm.

a) 40/9
b) 40/6
c) 4/3
d) 40/3 

Ans: d

Q 32) Sides of two similar triangles are in the ratio 4: 9. Areas of these triangles are in the ratio
(a) 2 : 3
(b) 4 : 9
(c) 81 : 16
(d) 16 : 81 

Ans: d

Q 33) The areas of two similar triangles are 100cm² and 64cm². If the altitude of the larger triangle is 5.5 cm, then what will be the altitude of the corresponding smaller triangle?

a) 4.4 cm 

b) 4.5 cm
c) 2.4 cm 

d) 2.5 cm
Ans:  a
Solution Hint ratio of areas of similar triangles is equal to the ratio of the squares of their corresponding altitudes.

Q 34) Which of the following are not similar figures?
(a) Circles
(b) Squares
(c) Equilateral triangles
(d) Isosceles triangles 

Ans: d

Q 35) The area of two similar triangles is 25cm² and 121cmx². If the median of the bigger triangle is 10 cm, then what will be the corresponding median of the smaller triangle?

a) 51 / 11
b) 10 / 11
c) 50 / 11
d) 5 / 11
Ans: c
Solution Hint ratio of areas of similar triangles is equal to the ratio of the squares of their corresponding medians.

Q 36) In triangle ABC, ∠BAC = 90° and AD ⊥ BC. Then
(a) BD . CD = BC²
(b) AB . AC = BC²
(c) BD . CD = AD²
(d) AB . AC = AD²

Ans: c

Q 37) ∆ABC ∼ ∆PQR, AD & PS are the angle bisectors of respectively. If AD = 1.5cm and PS = 2.3 cm then, what will be the ratio of the areas of ∆ABC and ∆PQR?
a) 19 : 15
b) 225 : 529
c) 529 : 225
d) 15 : 17
Ans: b
Solution Hint Areas of similar triangles is equal to the ratio of the squares of their corresponding angle bisectors.

Q 38) In triangles ABC and DEF, ∠B = ∠E, ∠F = ∠C and AB = 3 DE. Then, the two triangles are
(a) congruent but not similar
(b) similar but not congruent
(c) neither congruent nor similar
(d) congruent as well as similar 

Ans: b

Q 39) In the given figure, DE || BC and AE : EC = 3 : 4. What will be the ratio of the areas ∆ADE and ∆ABC?

a) 16 : 9        b) 7 : 3        c) 49 : 9        d) 9 : 49 

Ans: d

Q 40) It is given that ΔABC ~ ΔPQR, with BC / QR = 1/4 then, ar(ΔPRQ) / ar(ABC) is equal to
(a) 16
(b) 4
(c) 1/ 4
(d) 1/ 16
Ans: a
Solution Hint
Given, ΔABC ~ ΔPQR
and BC / QR = 1/4
Ratio of area of similar triangles is equal to the square of its corresponding sides. So, ar(ΔPRQ)/ar(ABC) = (QR/BC)² = (4/1)² = 16

Q 41)  What will be the length of the altitude of an equilateral triangle whose side is 8 cm?

a)  cm                  b)  cm            c)  cm                d)  cm

Ans: a

Q 42) It is given that ΔABC ~ ΔDFE, ∠A = 30°, ∠C = 50°, AB = 5 cm, AC = 8 cm and DF = 7.5 cm. Then, the following is true:

(a) DE = 12 cm, ∠F = 50° 

(b) DE = 12 cm, ∠F = 100°

(c) EF = 12 cm, ∠D = 100° 

(d) EF = 12 cm, ∠D = 30°

Ans: b

Solution Hint

Given, ΔABC ~ ΔDFE, ∠A =30°, ∠C = 50°, AB = 5 cm, AC = 8 cm and DF= 7.5 cm

In triangle ABC,

∠A + ∠B + ∠C = 180°

∠B = 180° – 30° – 50° = 100°

Since ΔABC ~ ΔDFE, the corresponding angles are equal.

Thus, ∠D = ∠A = 30°

∠F = ∠B = 100°

∠E = ∠C = 50°

And

AB / DF = AC / DE

5 / 7.5 = 8 / DE

DE = (8 × 7.5) / 5 = 12 cm

Q 43) In the given figure PQ || BC, and 4AP = 3AB. What will be the ratio of areas ∆ABC and trap.(PQCB)?

a) 9 : 7                   b) 7 : 9                     c) 16 : 7                       d) 16 : 9

Ans: c

Q 44) Which of the following is not a similarity criterion for two triangles?

(a) AAA                   (b) SAS                        (c) SSS                              (d) ASA

Ans: d

Q 45) The ratio of the areas of two similar triangles is equal to

(a) square of the ratio of their corresponding sides

(b) cube of the ratio of their corresponding sides

(c) square root of the ratio of their corresponding sides

(d) twice the ratio of their corresponding sides

Ans: a

Q 46) What will be the distance of the foot of ladder from the building, if the ladder is 10 m long reaches the top of a building 6 m high from the ground?

a) 16 m                 b) 8 m                    c) 12 m                         d) 15 m

Ans: b

Q 47) The heights of two vertical lamp posts are 11 m and 6 m high. If the distance between them is 12 m, then what will be the distance between their tops?

a) 15 m                    b) 5 m                    c) 23 m                        d) 13 m

Ans: d

Q 48) Which of triangle whose sides are given below are right angled?

a) AB = 3, AC = 8, BC = 6
b) AB = 5, AC = 12, BC = 15
c) AC = 7, AB = 24, BC = 25
d) AC = 7, AB = 24, BC = 26

Ans: c

Q 49) ∆ABC is a right-angled triangle, right angled at B and BD ⊥ AC. If BD = 5cm, AB = 12 cm and BC = 4 cm then AC will be?

a) 12 cm             b) 16 cm              c) 12.5 cm            d) 9.6 cm

Ans: d

Solution Hint

First Prove that ΔABC ~ ΔADB,  then corresponding sides are proportional

Therefore : 

  

Q 50) Diagonals of a rhombus are 15 cm and 36 cm long. Find the perimeter.

a) 96 cm                  b) 72 cm                 c) 78 cm                    d) 84 cm

Ans: c

Q 51) Sides of two similar triangles are in the ratio 4 : 5. If median of first triangle is 26 cm then find the corresponding  median of the second triangle.

a) 36 cm                    b) 28.5 cm                c) 32.5 cm                     d) 30 cm

Ans: c

Solution Hint:

When two triangles are similar then corresponding sides, medians, altitudes, angle bisectors and perimeters are proportional.

Q 52) If the side of rhombus is 13 cm and one of its diagonals is 24 cm, then what will be length of the other diagonal?

a) 8.4 cm                 b) 4 cm                   c) 11 cm                    d) 10 cm

Ans: d

Solution Hint: ABCD is a rhombus. The side of the rhombus is 13 cm and the length of one of its diagonal is 24 cm.

Let the length of other diagonal be 2x cm.
Since, diagonals of a rhombus bisect each other at right angle.

Therefore, AE = x cm and DE = 12 cm
Now, in ∆AED, by Pythagoras theorem
AD² = AE² + DE²
13² = x² + 12²     (Since, AD is the altitude of the triangle it will bisect BC)
x² = 169 – 144     ⇒  x² = 25
x = √25 = 5 cm = AE
AC = 2 × AE = 2 × 5 = 10 cm

Q 53) A ladder 17m long reaches the top of the building 15m high from the ground. Find the distance of the foot of the ladder from the building.

a)  8 m                b)  12 m            c)  7 m                     d) 6 m

Ans: a

Q 54) The lengths of diagonals of a rhombus are 10 cm and 8 cm. What will be the length of the sides of rhombus?

a) 6.40 cm                    b) 5.25 cm            c) 2.44 cm                        d) 3.29 cm

Ans: a

Solution Hint
Since, diagonals of a rhombus bisect each other at right angles.

Therefore, AE = 4cm and DE = 5 cm
Now, in ∆AED, by Pythagoras Theorem
AD² = AE² + DE²
AD² = 4² + 5 ²     (Since, AD is the altitude of the triangle it will bisect BC)
AD² = 16 + 25      ⇒     AD² = 41

AD = √41 cm = 6.40 cm

Q 55) A man travels A to B, B to C, C to D and then finally D to E. What will be the shortest route the man could have taken?

a) 13 m                        

b) 18 m                       

c) 15 m                     

d) 10 m

Ans: b

Questions From CBSE Sample Paper 2021-22

Q 56) In the given figure, DE II BC. Which of the following is true?

  

  

   

 

Ans: c

Q 57) The perimeters of two similar triangles are 26 cm and 39 cm. The ratio of their areas will be

(a) 2 : 3                 (b) 6 : 9        (c) 4 : 6                  (d) 4 : 9

Ans: d

Q 58) A vertical stick 20m long casts a shadow 10m long on the ground. At the same time a tower casts a shadow 50m long. What is the height of the tower?

(a) 30m                 (b) 50m            (c) 80m                  (d) 100m

Ans: d

Q 59) In an isosceles triangle ABC, if AC = BC and AB² = 2AC², then the measure of angle C will be

(a) 30 ̊                   (b) 45 ̊            (c) 60 ̊                     (d) 90 ̊

Ans: d

Q 60) A man goes 15m due west and then 8m due north. How far is he from the starting point?

(a) 7m                    (b) 10m            (c) 17m                  (d) 23m

Ans: c

Q 61) What is the length of an altitude of an equilateral triangle of side 8cm?

(a) 2√3 cm                                     

(b) 3√3 cm

(c) 4√3 cm                                    

(d) 5√3 cm

Ans: c

Q 62) In ∆ABC, ∠B = 90 ̊ and BD ꓕ AC. If AC = 9cm and AD = 3 cm then BD is equal to

(a) 2√2 cm                 (b) 3√2 cm            (c) 2√3 cm                   (d) 3√3 cm

Ans: b

Q 63) A girl walks 200m towards East and then 150m towards North. The distance of the girl from the starting point is

(a)350m                   (b) 250m            (c) 300m                  (d) 225

Ans: b

Q 64) The lengths of the diagonals of a rhombus are 24cm and 32cm, then the length of the

altitude of the rhombus is

(a) 12cm                  (b) 12.8cm            (c) 19 cm             (d) 19.2cm

Ans: d

Q 65) ∆ABC ~ ∆PQR. If AM and PN are altitudes of ∆ABC and ∆PQR respectively and AB² : PQ² = 4 : 9, then AM : PN =

(a) 16:81                (b) 4:9            (c) 3:2                       (d) 2:3

Ans: d

Q 66) ∆ABC is such that AB = 3 cm, BC =  2cm, CA = 2.5 cm. If ∆ABC ~ ∆DEF and EF = 4cm, then perimeter of ∆DEF is

(a) 7.5 cm               (b) 15 cm                (c) 22.5 cm              (d) 30 cm

Ans: b

Q 67) In the figure, if DE|| BC, AD = 3cm, BD = 4cm and BC = 14 cm, then DE equals

(a) 7cm                (b) 6cm                        (c) 4cm                             (d) 3cm

Ans: b

Q 68) In the given figure, ∠ACB = ∠CDA, AC = 8cm, AD = 3cm, then BD is

(a) 22/3 cm               (b) 26/3 cm            (c) 55/3 cm                 (d) 64/3 cm

Ans: c

Q 69) Sides AB and BE of a right triangle, right angled at B are of lengths 16 cm and 8 cm respectively. The length of the side of largest square FDGB that can be inscribed in the triangle ABE is

a) 32/3cm              (b) 16/3cm             (c)8/3cm                    (d) 4/3cm

Ans: b

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