Triangles: Class 10 Important Questions with Answers

HOTS (higher-order) practice

H1. In △ABC, DE ∥ BC with D on AB, E on AC. If AD = x, DB = x − 2, AE = x + 2 and EC = x − 1, find x.

H2. A 1.8 m tall man stands 6 m from a lamp post and casts a 3 m shadow. How tall is the lamp post?

H3. In trapezium ABCD, AB ∥ DC and AB = 2CD. If the diagonals meet at O, find AO : OC.

Case-based practice

No 2023–2025 Standard paper set a full case study on this chapter, so these two follow the board's case format with the ideas it does test.

20 MCQs (1 mark each)

1. Which are always similar?
   (a) two rectangles (b) two rhombuses (c) two circles (d) two isosceles triangles
2. In △ABC, DE ∥ BC, AD = 3, DB = 6, AE = 2. Then EC is
   (a) 4 (b) 3 (c) 6 (d) 1
3. If △ABC ~ △PQR, then $\frac{AB}{PQ}$ equals
   (a) $\frac{BC}{PR}$ (b) $\frac{CA}{RP}$ (c) $\frac{AC}{PQ}$ (d) $\frac{BC}{PQ}$
4. DE ∥ BC with AD = 2, AB = 6, BC = 9. Then DE is
   (a) 3 (b) 4.5 (c) 6 (d) 2
5. Which is not a similarity criterion?
   (a) AA (b) SSS (c) SAS (d) RHS
6. Two similar triangles have sides in ratio 2 : 3. Their perimeters are in ratio
   (a) 4 : 9 (b) 2 : 3 (c) 3 : 2 (d) 8 : 27
7. In △ABC, ∠A = 50°, ∠B = 60°; in △PQR, ∠P = 50°, ∠R = 70°. Then
   (a) △ABC ~ △PQR (b) △ABC ~ △PRQ (c) not similar (d) △ABC ~ △QPR
8. The line joining mid-points of two sides of a triangle is
   (a) perpendicular to the third side (b) parallel to the third side (c) equal to the third side (d) a median
9. A 2 m stick casts a 3 m shadow when a pole casts a 12 m shadow. The pole is
   (a) 8 m (b) 18 m (c) 6 m (d) 9 m
10. In trapezium ABCD, AB ∥ DC, AB = 9, DC = 3. Diagonals meet at O. AO : OC is
    (a) 1 : 3 (b) 3 : 1 (c) 9 : 1 (d) 1 : 1
11. If $\frac{AD}{DB}=\frac{AE}{EC}$ in △ABC, then
    (a) DE ⊥ BC (b) DE ∥ BC (c) DE = BC (d) DE = ½BC
12. △ABC ~ △DEF, AB = 4, DE = 6, EF = 9. BC is
    (a) 6 (b) 13.5 (c) 4 (d) 8
13. In △ABC right-angled at A, AD ⊥ BC, BD = 9, DC = 4. AD is
    (a) 6 (b) 13 (c) 36 (d) 5
14. Two triangles are similar if their corresponding
    (a) angles are proportional (b) sides are equal (c) angles are equal (d) none
15. D on BC with ∠ADC = ∠BAC, CB = 9, CD = 4. CA is
    (a) 6 (b) 36 (c) 13 (d) 5
16. DE ∥ BC, AD = x, DB = x − 2, AE = x + 2, EC = x − 1. x is
    (a) 2 (b) 4 (c) 3 (d) 1
17. If △ABC ~ △PQR with ratio 1 : 1, the triangles are
    (a) congruent (b) not similar (c) right-angled (d) equilateral
18. A square of side 4 and a rectangle 4 × 6 are
    (a) similar (b) congruent (c) not similar (d) both
19. In △ABC, D, E are mid-points of AB, AC and BC = 10 cm. DE is
    (a) 10 cm (b) 20 cm (c) 5 cm (d) 2.5 cm
20. △ABC ~ △QRP. The side corresponding to BC is
    (a) QR (b) RP (c) PQ (d) QP

Answer key

1 (c) · 2 (a) · 3 (b) · 4 (a) · 5 (d) · 6 (b) · 7 (a) · 8 (b) · 9 (a) · 10 (b) · 11 (b) · 12 (a) · 13 (a) · 14 (c) · 15 (a) · 16 (b) · 17 (a) · 18 (c) · 19 (c) · 20 (b)

Hints: Q4 $\frac{DE}{9}=\frac{2}{6}$. Q7 ∠C = 70° = ∠R, ∠B = ∠Q = 60°. Q13 $A{D}^2=9\times 4$. Q15 $C{A}^2=9\times 4$. Q19 mid-point theorem: $DE=\frac{1}{2}BC$. Q20 B↔R, C↔P.

Assertion–Reason (10)

(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.

1. A: All equilateral triangles are similar. R: All their angles are 60°, so AA holds.
2. A: A square and a rhombus of equal side are similar. R: Their sides are proportional.
3. A: In △ABC, DE ∥ BC with AD = 2, DB = 4, AE = 3; then EC = 6. R: A line parallel to one side divides the other two sides in the same ratio.
4. A: Two congruent triangles are similar. R: Two similar triangles are always congruent.
5. A: If ∠A = ∠P and ∠B = ∠Q, then △ABC ~ △PQR. R: If two angles match, the third angles also match.
6. A: △ABC with sides 3, 4, 5 is similar to △PQR with sides 6, 8, 10. R: The sides are in the same ratio 1 : 2.
7. A: If $\frac{AB}{PQ}=\frac{AC}{PR}$ and ∠B = ∠Q, then △ABC ~ △PQR. R: In SAS similarity the angle must be included between the two sides.
8. A: The ratio of the perimeters of two similar triangles equals the ratio of their corresponding sides. R: Each side is multiplied by the same number $k$.
9. A: Any two right triangles are similar. R: Each has an angle of 90°.
10. A: In trapezium ABCD with AB ∥ DC, △AOB ~ △COD. R: Vertically opposite angles are equal.

Key: 1 (a) · 2 (d) · 3 (a) · 4 (c) · 5 (a) · 6 (a) · 7 (d) · 8 (a) · 9 (d) · 10 (b)

Q7: ∠B is not between AB and AC, so A is false. Q10: A is true, but it needs the alternate angles from AB ∥ DC as well; R alone is not enough ⇒ (b).

Very short answer (1 mark each)

1. State the converse of BPT. Ans. A line dividing two sides in the same ratio is parallel to the third side.
2. In △ABC, DE ∥ BC, AD = 4, AB = 10, AC = 15. Find AE. Ans. 6
3. △ABC ~ △PQR and ∠A = 40°, ∠Q = 75°. Find ∠R. Ans. 65°
4. Sides of two similar triangles are 6 and 9; the first perimeter is 24. Find the second. Ans. 36
5. Write the criterion: $\frac{AB}{PQ}=\frac{BC}{QR}=\frac{CA}{RP}$. Ans. SSS
6. Are all squares similar? Why? Ans. Yes: angles all 90°, sides in one ratio.
7. In △ABC, AD ⊥ BC, ∠A = 90°, BD = 8, DC = 2. Find AD. Ans. 4
8. A 3 m pole casts a 2 m shadow. Find the shadow of a 12 m tree at the same time. Ans. 8 m

Short answer practice (2 marks each)

1. In △ABC, DE ∥ BC. If $\frac{AD}{DB}=\frac{3}{5}$ and AC = 5.6 cm, find AE.

2. In the figure (bow-tie), AB ∥ CD and AD, BC meet at O. Show that $\frac{OA}{OD}=\frac{OB}{OC}$.

3. △ABC ~ △DEF, BC = 3 cm, EF = 4 cm and the perimeter of △ABC is 9 cm. Find the perimeter of △DEF.

Short answer practice (3 marks each)

4. In △PQR, S and T are points on PQ and PR with ST ∥ QR. If PS = x, SQ = x − 2, PT = x + 2 and TR = x − 1, find x.

5. In △ABC, AD is the bisector of ∠A and DE ∥ AC meets AB at E. Prove that △BED ~ △BAC and hence $\frac{BE}{BA}=\frac{ED}{AC}$.

6. In △ABC, P and Q are on AB and AC with AP = 2, PB = 4, AQ = 3, QC = 6. Show that PQ = $\frac{1}{3}$BC.

Long answer (5 marks each)

7. State and prove the Basic Proportionality Theorem. Using it, prove that in trapezium ABCD (AB ∥ DC), if EF ∥ AB meets AD at E and BC at F, then $\frac{AE}{ED}=\frac{BF}{FC}$.

8. In △ABC, D is a point on BC such that ∠ADC = ∠BAC. If BC = 16 cm and CD = 4 cm, find AC. Also, if AB = 10 cm, find AD.