Class 10 Maths Chapter 6 Triangles MCQ / Objective Questions with Answers in English

Contents hide
1 Direct Questions
2 Assertion-Reason
3 Match the Following
4 Statement-Based Questions
5 Fill in the Blanks
6 Odd One Out
7 Passage-Based Questions
8 True / False

20 multiple-choice (MCQ / objective) questions from this chapter, in the common exam-style shapes. Every question comes with an explanation along with the answer, so you remember why the option is correct, not just which one it is.

1 Direct Questions

Direct Questions
  1. 1 markIn △ABC\triangle ABC, DE∥BCDE \parallel BC, AD=xAD=x, DB=x−2DB=x-2, AE=x+2AE=x+2 and EC=x−1EC=x-1. The value of xx is:
    (A) 2(B) 3(C) 4(D) 6
  2. 1 markIn △ABC\triangle ABC, DE∥BCDE \parallel BC and AD:DB=3:5AD:DB=3:5. If AC=5.6AC=5.6 cm, then AE is:
    (A) 2.1 cm(B) 3.5 cm(C) 3.36 cm(D) 1.8 cm
  3. 1 markA vertical pole of length 6 m casts a shadow 4 m long, and at the same time a tower casts a shadow 28 m long. The height of the tower is:
    (A) 42 m(B) 18.67 m(C) 36 m(D) 48 m
  4. 1 markD is a point on side BC of △ABC\triangle ABC such that ∠ADC=∠BAC\angle ADC=\angle BAC. If CB=9CB=9 cm and CD=4CD=4 cm, then CA is:
    (A) 36 cm(B) 6 cm(C) 13 cm(D) 5 cm
  5. 1 markABCD is a trapezium with AB∥DCAB \parallel DC; its diagonals meet at O. If OA=6OA=6 cm, OC=9OC=9 cm and OB=4OB=4 cm, then OD is:
    (A) 2.67 cm(B) 8 cm(C) 9 cm(D) 6 cm
  6. 1 markA lamp is 5 m above the ground. A boy 1.25 m tall stands 6 m from the foot of the lamp-post. The length of his shadow is:
    (A) 2 m(B) 1.5 m(C) 24 m(D) 3 m
  7. 1 markD and E are points on sides AB and AC of △ABC\triangle ABC. For which set of lengths (in cm) is DE∥BCDE \parallel BC?
    (A) AD=4AD=4, DB=6DB=6, AE=5AE=5, EC=8EC=8(B) AD=6AD=6, DB=9DB=9, AE=8AE=8, EC=12EC=12(C) AD=3AD=3, DB=4DB=4, AE=4AE=4, EC=3EC=3(D) AD=5AD=5, DB=10DB=10, AE=4AE=4, EC=6EC=6
  8. 1 markLines AC and BD meet at O, △ODC∼△OBA\triangle ODC \sim \triangle OBA, ∠BOC=115∘\angle BOC=115^\circ and ∠CDO=80∘\angle CDO=80^\circ. Then ∠OAB\angle OAB is:
    (A) 35∘35^\circ(B) 65∘65^\circ(C) 80∘80^\circ(D) 115∘115^\circ
  9. 1 markWhich pair of figures is ALWAYS similar?
    (A) Two isosceles triangles(B) Two rhombuses(C) Two rectangles(D) Two equilateral triangles
  10. 1 markIn △ABC\triangle ABC and △PQR\triangle PQR, ∠A=∠Q\angle A=\angle Q, ∠B=∠R\angle B=\angle R and ∠C=∠P\angle C=\angle P. The correct similarity statement is:
    (A) △ABC∼△PQR\triangle ABC \sim \triangle PQR(B) △ABC∼△QRP\triangle ABC \sim \triangle QRP(C) △ABC∼△RPQ\triangle ABC \sim \triangle RPQ(D) △ABC∼△PRQ\triangle ABC \sim \triangle PRQ

2 Assertion-Reason

Assertion-Reason
  1. 1 markAssertion (A): All congruent polygons are similar.
    Reason (R): Congruent polygons have all corresponding angles equal and all corresponding sides in the ratio 1 : 1.
    (A) Both A and R are true, and R is the correct explanation of A(B) Both A and R are true, but R is not the correct explanation of A(C) A is true, but R is false(D) A is false, but R is true
  2. 1 markAssertion (A): △ABC\triangle ABC with AB=3AB=3 cm, ∠A=50∘\angle A=50^\circ, AC=4AC=4 cm is similar to △DEF\triangle DEF with DE=6DE=6 cm, ∠E=50∘\angle E=50^\circ, DF=8DF=8 cm.
    Reason (R): In the SAS criterion, the equal angles must be included between the proportional sides.
    (A) Both A and R are true, and R is the correct explanation of A(B) Both A and R are true, but R is not the correct explanation of A(C) A is true, but R is false(D) A is false, but R is true

3 Match the Following

Match the Following
  1. 1 markMatch the criterion in Column A with what it needs in Column B.
    Column A: (i) AA (ii) SSS (iii) SAS (iv) RHS
    Column B: (a) all three pairs of sides proportional (b) right triangles with hypotenuse and one side proportional (c) two pairs of angles equal (d) one angle equal and the sides including it proportional
    (A) i-a, ii-c, iii-d, iv-b(B) i-c, ii-d, iii-a, iv-b(C) i-c, ii-a, iii-d, iv-b(D) i-b, ii-a, iii-d, iv-c

4 Statement-Based Questions

Statement-Based Questions
  1. 1 markStatement I: All circles are similar.
    Statement II: All squares are congruent.
    (A) Both Statement I and Statement II are true(B) Both Statement I and Statement II are false(C) Statement I is true, but Statement II is false(D) Statement I is false, but Statement II is true
  2. 1 markStatement I: A square and a rhombus whose sides are in the same ratio are always similar.
    Statement II: The scale factor of two similar polygons is also called the Representative Fraction.
    (A) Both Statement I and Statement II are true(B) Both Statement I and Statement II are false(C) Statement I is true, but Statement II is false(D) Statement I is false, but Statement II is true

5 Fill in the Blanks

Fill in the Blanks
  1. 1 markThe symbol used for ‘is similar to’ is ______.
    (A) ≅\cong(B) ∼\sim(C) ==(D) ∥\parallel
  2. 1 markIf △ABC∼△PQR\triangle ABC \sim \triangle PQR, ABPQ=35\frac{AB}{PQ}=\frac35 and QR=15QR=15 cm, then BC=BC= ______.
    (A) 25 cm(B) 12 cm(C) 9 cm(D) 6 cm

6 Odd One Out

Odd One Out
  1. 1 markWhich of these does NOT guarantee that two triangles are similar?
    (A) Two angles of one equal to two angles of the other(B) All three sides proportional(C) Two sides proportional and a non-included angle equal(D) Two sides proportional and the included angle equal

7 Passage-Based Questions

Passage-Based Questions
Passage
A ramp is built from a point B on level ground up to a point A on a platform. The platform edge AC is vertical, 3 m high, and C is on the ground 6 m from B (so BC = 6 m). Vertical support posts are fixed under the ramp: each post stands on BC and reaches up to the ramp AB.
  1. 1 markA post DE stands at D on BC with BD=2BD=2 m, and its top E is on the ramp. The height DE is:
    (A) 2 m(B) 1.5 m(C) 1 m(D) 0.5 m

8 True / False

True / False
  1. 1 markP: All congruent figures are similar. Q: All similar figures are congruent. R: Any two equilateral triangles are similar. Mark each True (T) or False (F), in the order P, Q, R.
    (A) T T T(B) T F T(C) F T T(D) T F F
Answer Key and Explanations
Direct Questions
  1. (C) xx−2=x+2x−1\frac{x}{x-2}=\frac{x+2}{x-1} gives x2−x=x2−4x^2-x=x^2-4, so x=4x=4. Check: AD=4AD=4, DB=2DB=2, AE=6AE=6, EC=3EC=3, and 42=63\frac{4}{2}=\frac{6}{3}.
  2. (A) AE:EC=3:5AE:EC=3:5, so AE=38×5.6=2.1AE=\frac{3}{8}\times5.6=2.1 cm.
  3. (A) Pole and tower make similar triangles with their shadows (AA). h28=64\frac{h}{28}=\frac{6}{4}, so h=42h=42 m.
  4. (B) △ADC∼△BAC\triangle ADC \sim \triangle BAC (AA: ∠ADC=∠BAC\angle ADC=\angle BAC, ∠C\angle C common), so CACB=CDCA\frac{CA}{CB}=\frac{CD}{CA}, i.e. CA2=CB⋅CD=36CA^2=CB\cdot CD=36 and CA=6CA=6 cm.
  5. (D) △OAB∼△OCD\triangle OAB \sim \triangle OCD (AA, alternate angles), so OAOC=OBOD\frac{OA}{OC}=\frac{OB}{OD}: 69=4OD\frac{6}{9}=\frac{4}{OD}, OD=6OD=6 cm.
  6. (A) By AA, 6+xx=51.25=4\frac{6+x}{x}=\frac{5}{1.25}=4, so 6+x=4x6+x=4x and x=2x=2 m.
  7. (B) Only in (B) are the ratios equal: 69=23=812\frac69=\frac23=\frac{8}{12}. In (A) 46≠58\frac46\ne\frac58, in (C) 34≠43\frac34\ne\frac43, in (D) 12≠23\frac12\ne\frac23.
  8. (A) ∠DOC=180∘−115∘=65∘\angle DOC=180^\circ-115^\circ=65^\circ (linear pair). ∠DCO=180∘−80∘−65∘=35∘\angle DCO=180^\circ-80^\circ-65^\circ=35^\circ. From the similarity, ∠OAB=∠OCD=35∘\angle OAB=\angle OCD=35^\circ.
  9. (D) All equilateral triangles have every angle 60∘60^\circ and all sides of each one equal, so the side ratios always match. Isosceles triangles, rhombuses and rectangles can change shape.
  10. (B) Write the vertices in corresponding order: A matches Q, B matches R, C matches P, so ABC is similar to QRP.
Assertion-Reason
  1. (A) Both are true, and R explains A: two polygons are similar when their corresponding angles are equal and their corresponding sides are in the same ratio, and a ratio of 1 : 1 is such a ratio.
  2. (D) A is false. In DEF the 50∘50^\circ angle is at E, not between DE and DF, so SAS cannot be used. In fact the triangles are not similar: DEF is fixed by DE=6DE=6, ∠E=50∘\angle E=50^\circ, DF=8DF=8 and its angles are 50∘50^\circ, about 35∘35^\circ (at F) and about 95∘95^\circ, while the angles of ABC are 50∘50^\circ, about 82∘82^\circ (at B) and about 48∘48^\circ. R is true.
Match the Following
  1. (C) AA needs two equal angles (c), SSS needs three proportional sides (a), SAS needs an included angle with proportional sides (d), and RHS is for right triangles (b).
Statement-Based Questions
  1. (C) Statement I is true: every circle has the same shape. Statement II is false: squares are congruent only when their sides are equal.
  2. (D) Statement I is false: the angles of a rhombus need not be 90∘90^\circ, so the angles may not be equal. Statement II is true, as the book states.
Fill in the Blanks
  1. (B) ∼\sim means ‘is similar to’; ≅\cong means ‘is congruent to’.
  2. (C) BCQR=ABPQ=35\frac{BC}{QR}=\frac{AB}{PQ}=\frac35, so BC=35×15=9BC=\frac35\times15=9 cm.
Odd One Out
  1. (C) The other three are AA, SSS and SAS. A non-included angle with two proportional sides is not a similarity criterion.
Passage-Based Questions
  1. (C) △BDE∼△BCA\triangle BDE \sim \triangle BCA (AA: ∠B\angle B common, ∠BDE=∠BCA=90∘\angle BDE=\angle BCA=90^\circ). So DECA=BDBC\frac{DE}{CA}=\frac{BD}{BC} and DE=3×26=1DE=\frac{3\times2}{6}=1 m.
True / False
  1. (B) P is true; Q is false (similar figures may differ in size); R is true.
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