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Triangles

Mathematics · Class 10 · All

MathematicsClass 10CBSEEasy
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Right-angled isosceles triangle at B, AP perp bisector of BC. AC² =

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All 110 questions in this quiz

  1. Which statement best summarizes the chapter "Triangles"?

    1. Similar triangles always have equal sides.
    2. Only right triangles can be similar.
    3. The chapter explains similarity criteria, proportionality theorems, relationships between corresponding sides and areas, and the Pythagoras Theorem, which are essential for solving geometric problems.
    4. Every triangle is similar to every other triangle.

  2. Which of the following is NOT a similarity criterion?

    1. AA
    2. SAS
    3. SSS
    4. ASA

  3. If a line divides two sides of a triangle proportionally, then according to the converse of BPT, the line is:

    1. Perpendicular to the third side
    2. Parallel to the third side
    3. Equal to the third side
    4. Bisects the third side

  4. Which statement about similar triangles is true?

    1. Corresponding angles are unequal.
    2. Corresponding sides are proportional.
    3. All sides are equal.
    4. Areas are always equal.

  5. If AB/DE = BC/EF = AC/DF, then the triangles are:

    1. Congruent
    2. Similar by SSS criterion
    3. Right triangles only
    4. Isosceles triangles

  6. Which theorem is used in a right triangle?

    1. Midpoint Theorem
    2. Basic Proportionality Theorem
    3. Pythagoras Theorem
    4. Angle Bisector Theorem

  7. In two similar triangles, if one side is doubled, then every corresponding side is:

    1. Halved
    2. Doubled
    3. Tripled
    4. Unchanged

  8. If two similar triangles have corresponding sides in the ratio 3:5, then the ratio of their areas is:

    1. 3:5
    2. 6:10
    3. 9:25
    4. 15:25

  9. Which similarity criterion requires one angle to be equal and the including sides proportional?

    1. AA
    2. SAS
    3. SSS
    4. RHS

  10. If the areas of two similar triangles are in the ratio 25:49, then the ratio of their corresponding sides is:

    1. 25:49
    2. 5:7
    3. 7:5
    4. 10:14

  11. Which theorem states that a line drawn parallel to one side of a triangle divides the other two sides proportionally?

    1. Pythagoras Theorem
    2. Basic Proportionality Theorem (BPT)
    3. Midpoint Theorem
    4. Angle Bisector Theorem

  12. If two triangles are similar, the ratio of their corresponding sides is:

    1. Equal
    2. Zero
    3. Infinite
    4. Always 1

  13. If two triangles are similar, then their corresponding angles are:

    1. Supplementary
    2. Equal
    3. Complementary
    4. Unequal

  14. Which criterion is known as the Side-Side-Side (SSS) similarity criterion?

    1. Corresponding angles are equal.
    2. Corresponding sides are proportional.
    3. One side is equal.
    4. Only one angle is equal.

  15. The corresponding sides of two similar triangles are in the ratio 7 : 9. If the perimeter of the smaller triangle is 63 cm, what is the perimeter of the larger triangle?

    1. 72 cm
    2. 81 cm
    3. 84 cm
    4. 90 cm

  16. In triangle ABC, D and E are points on AB and AC respectively such that DE is parallel to BC. If AD = 5 cm, AB = 15 cm and AC = 18 cm, then find AE.

    1. 4 cm
    2. 5 cm
    3. 6 cm
    4. 8 cm

  17. Triangles ABC and PQR are similar. If the ratio of their corresponding sides is 3 : 5 and the area of triangle PQR is 200 cm², then what is the area of triangle ABC?

    1. 72 cm²
    2. 80 cm²
    3. 90 cm²
    4. 120 cm²

  18. In triangle ABC, point D lies on BC such that BD : DC = 2 : 3. If the area of triangle ABD is 48 cm², find the area of triangle ADC.

    1. 60 cm²
    2. 72 cm²
    3. 75 cm²
    4. 80 cm²

  19. Two similar triangles have corresponding sides in the ratio 4 : 7. If the difference between their perimeters is 39 cm, find the perimeter of the larger triangle.

    1. 91 cm
    2. 84 cm
    3. 78 cm
    4. 98 cm

  20. In triangle ABC, points D and E lie on sides AB and AC respectively such that DE is parallel to BC. If AD = 6 cm, DB = 3 cm and AE = 8 cm, then find the length of EC.

    1. 2 cm
    2. 3 cm
    3. 4 cm
    4. 6 cm

  21. If Σ(3/4)ˣ from x=0 to ∞ is finite, then

    1. x<4
    2. x>4
    3. x<1/4
    4. x>1/4

  22. If 1st, 2nd and middle terms of AP are a, b, c, sum of series is

    1. 2c(c–a)/(b–a)
    2. 2c(b–a)/(c–a)
    3. 2c(c–a)/(b–a)
    4. c(c–a)/(b–a)

  23. Ratio of sum of m and n terms of AP is m²:n². Ratio of mth and nth terms is

    1. m–1:n–1
    2. 2m+1:2n+1
    3. 2m–1:2n–1
    4. m:n

  24. If a₁,a₂,...,aₙ are in AP, sum of 1/√(a₁+√a₂)+... equals

    1. (n–1)/√(a₁+aₙ)
    2. n/√(a₁+aₙ)
    3. (n–1)/√(a₁·aₙ)
    4. none

  25. If AM of roots of quadratic is 8/5 and AM of reciprocals is 8/7, the quadratic is

    1. 5x²–8x+7=0
    2. 5x²–16x+7=0
    3. 7x²–16x+5=0
    4. 7x²+16x+5=0

  26. If a,b,c are in AP, which is NOT true?

    1. 1/a,1/b,1/c are in HP
    2. a+k,b+k,c+k are in AP
    3. a²,b²,c² are in AP
    4. a+b,c+a,b+c are in AP

  27. 7th term of AP is 40. Sum of first 13 terms is

    1. 520
    2. 53
    3. 2080
    4. 1040

  28. 5th term of GP is 2; product of first 9 terms is

    1. 256
    2. 128
    3. 18
    4. 512

  29. Sum of infinite GP is 20 and sum of squares is 100. Common ratio is

    1. 5
    2. 1/2
    3. 3/4
    4. 2/3

  30. If nth terms of APs 3,10,17,... and 63,65,67,... are equal, then n =

    1. 13
    2. 15
    3. 9
    4. 8

  31. If AM of roots of quadratic is 8 and GM is 5, the equation is

    1. x²+16x–25=0
    2. x²–8x+5=0
    3. x²–16x+25=0
    4. x²–16x–25=0

  32. In AP of 81 terms, 41st term is 10. Sum of series is

    1. 41×41
    2. 10×81
    3. 10×41
    4. 81×10/2

  33. Value of (7+77+777+...)/7 is

    1. 3
    2. 6
    3. 9
    4. 1

  34. If pth, qth, rth terms of GP are in GP, then p,q,r are in

    1. AP
    2. GP
    3. HP
    4. p=q=r

  35. If aˣ=bʸ=c^z and a,b,c are in GP, then x,y,z are in

    1. AP
    2. GP
    3. HP
    4. AGP

  36. If p,q,r are in AP and x,y,z are in GP, then x^(q–r)·y^(r–p)·z^(p–q) =

    1. 1
    2. xyz
    3. x+y+z
    4. p+q+r

  37. If p,q,r are in AP, then pth, qth, rth terms of any GP are in

    1. AP
    2. GP
    3. HP
    4. AGP

  38. If Sᵣ denotes sum of first r terms of GP, then Sₙ, S₂ₙ–Sₙ, S₃ₙ–S₂ₙ are in

    1. AP
    2. GP
    3. HP
    4. AGP

  39. If a>0,b>0,c>0 are in GP, logₐx, logᵦx, log_cx are in

    1. AP
    2. GP
    3. HP
    4. AGP

  40. If A is single AM and S is sum of n AMs between a and b, then S/A depends on

    1. n,a,b
    2. n,a
    3. n,b
    4. n

  41. If a,b,c are in AP, then 10^(ax+10), 10^(bx+10), 10^(cx+10) are in

    1. AP
    2. GP only when x>0
    3. GP for all x
    4. GP only when x<0

  42. Third term of GP is 4. Product of first five terms is

    1. 4³
    2. 4⁵
    3. 4⁴
    4. 4⁶

  43. 20th term of series 2×4+4×6+6×8+... is

    1. 1600
    2. 1680
    3. 420
    4. 840

  44. Sum of n terms of AP is Pn+Qn². Common difference is

    1. P+Q
    2. P–Q
    3. 2P
    4. 2Q

  45. First term of AP is p²+1, sum of (2p+1) terms can be expressed as

    1. (p+1)²
    2. (2p+1)(p+1)²
    3. (p+1)³
    4. p³+(p+1)³

  46. AP with n (odd) terms and middle term m. Sum of AP is

    1. 2mn
    2. mn/2
    3. mn
    4. mn²

  47. If a,b,c,d,e form AP, then a–4b+6c–4d+e =

    1. 1
    2. 2
    3. 0
    4. 3

  48. If a,b,c,d,e,f are in AP, then e–c equals

    1. 2(c–a)
    2. 2(f–d)
    3. 2(d–c)
    4. d–c

  49. If 3rd term of AP is 12 and 7th term is 24, then 10th term is

    1. 36
    2. 39
    3. 30
    4. 33

  50. If a, b, c are in AP, then 3ᵃ, 3ᵇ, 3ᶜ are in

    1. AP
    2. GP
    3. HP
    4. AGP

  51. For AP with first term a and common difference d, the nth term is

    1. aₙ=a+nd
    2. aₙ=a–(n+1)d
    3. aₙ=2a+(n–1)d
    4. aₙ=a+(n–1)d

  52. If a=–1, d=1/2, the AP terms are

    1. –1, 4, ...
    2. –1, –1/2, 0, 1/2,...
    3. –1, –1/2, 0, 1/2,...
    4. –1, 4,...

  53. First five terms of the sequence aₙ = (n–1)/(n+1) are

    1. 0,1/3,1/2,3/5,2/3
    2. 0,1/3,2/4,3/5,4/6
    3. 1/2,2/3,3/4,4/5,5/6
    4. 1,2,3,4,5

  54. If sums of n, 2n, 3n terms of AP are S₁, S₂, S₃, then S₃/(S₂–S₁) =

    1. 0
    2. 1/2
    3. 2
    4. 3

  55. Sum of all two-digit natural numbers is

    1. 4900
    2. 4500
    3. 4905
    4. 4599

  56. If x, a₁, a₂, y and x, b₁, b₂, y are each in AP, then (a₁–a₂)/(b₁–b₂) =

    1. 0
    2. –1
    3. 2
    4. 1

  57. AM between 10½ and 25½ is

    1. 17.9
    2. 18
    3. 19.5
    4. 18.5

  58. Sum of series 45²–43²+44²–42²+43²–41²+... to 30 terms is

    1. 2000
    2. 1790
    3. 2210
    4. 2220

  59. If 'a' is AM of 3 numbers and 'b' is AM of their squares, the AM of pairwise products in terms of a and b is

    1. (9a²–b)/3
    2. (9a²–b)/6
    3. (b–3a²)/2
    4. (9a²–b)/2

  60. Sum of first 3000 positive integers is

    1. 450150
    2. 4051500
    3. 4501500
    4. 4508542

  61. Sum of first 14 terms of an AP is 1050 and first term is 10. The 20th term is

    1. 100
    2. 300
    3. 200
    4. 700

  62. Sum of first 50 terms of AP: 8, 3, –2,...

    1. –2225
    2. 2225
    3. 2220
    4. –5725

  63. Flower bed rows: 23, 19, 15,..., 7 rose plants. How many rows?

    1. 4
    2. 3
    3. 5
    4. 7

  64. How many two-digit numbers are divisible by 5?

    1. 15
    2. 17
    3. 18
    4. 20

  65. What is the sum of first 54 terms of 5, 11, 17, 23,...?

    1. 287
    2. 8856
    3. 5871
    4. 4856

  66. In an AP, 3rd term is 11 and 7th term is 29. The AP starting values are

    1. 5, 9, 13,...
    2. 2, 6.5, 11,...
    3. 2, 6, 10,...
    4. 3, 7, 11,...

  67. Which term of the AP: 21, 18, 15,... is –81?

    1. 36
    2. 33
    3. 37
    4. 35

  68. What is the 10th term of the AP: 3, 10, 17,...?

    1. 56
    2. 46
    3. 17
    4. 66

  69. Which of the following lists of numbers form an AP? (i) 4,10,16,22,... (ii) 1,–1,–3,–5,... (iii) –2,2,–2,2,... (iv) 1,1,1,2,2,2,...

    1. i, ii & iv
    2. ii, iii & iv
    3. i & ii
    4. iii & iv

  70. For the AP: 3/2, 1/2, –1/2, –3/2,..., the common difference d is

    1. d=1
    2. d=–1
    3. d=2
    4. d=5

  71. Right triangle ABC at B, medians AD and CE from A and C. AC=5cm, AD=3√5/2. CE =

    1. √5 cm
    2. 2√5 cm
    3. 2 cm
    4. 2 cm

  72. Similar triangles areas 121 cm² and 64 cm². Median of first is 12.1 cm. Corresponding median of other is

    1. 8 cm
    2. 8.5 cm
    3. 8.9 cm
    4. 8.7 cm

  73. Right-angled isosceles triangle at B, AP perp bisector of BC. AC² =

    1. AP²+2BP²
    2. AP²–2BP²
    3. AP²+2BP·CP
    4. AP²–2BP·CP

  74. AB||MN, PA=x–2, PM=x, PB=x–1, PN=x+2. Value of x is

    1. 2
    2. 3
    3. 4
    4. 5

  75. Ladder foot is 5.5m from wall, top reaches 9m high. Length of ladder is

    1. 10.52 m
    2. 10.54 m
    3. 11 m
    4. 10.9 m

  76. Girl 120cm tall walks from lamp (3.6m) at 1.2m/s. Shadow after 4 seconds is

    1. 1.6 m
    2. 2.2 m
    3. 2.4 m
    4. 2.6 m

  77. Triangle ABC~PQR, angle A=80°, angle B=70°. Find angle R.

    1. 20°
    2. 30°
    3. 35°
    4. 37°

  78. In triangle ABC, AB>AC and AD⊥BC. AB²–AC² =

    1. BD²+AD²
    2. BD²+CD²
    3. BD²+AC²
    4. BD²–CD²

  79. Rhombus ABCD. AB²+BC²+CD²+DA² =

    1. OA²+OB²
    2. OB²+OC²
    3. OC²+OD²
    4. AC²+BD²

  80. In right triangle ABC right angled at C, AD is median. AB² =

    1. AC²–AD²
    2. AD²–AC²
    3. 3AC²–4AD²
    4. 4AD²–3AC²

  81. CD⊥AB, CD=p. Then 1/p² =

    1. 1/a²+1/b²
    2. 1/a²–1/b²
    3. 1/a·b
    4. 1/(a+b)

  82. ABC obtuse. AC² =

    1. AB²+BC²–2BC·BD
    2. AB²+BC²
    3. AB²+BC²+2BC·BD
    4. AD²+BD²

  83. Areas of similar triangles 81:49. Ratio of angle-bisector segments is

    1. 5:4
    2. 9:7
    3. 4:5
    4. 625:256

  84. P, Q on AB, AC of triangle ABC: AP=3.5, PB=7, AQ=3, QC=6, PQ=4.5. BC=

    1. 11.5 cm
    2. 13.5 cm
    3. 14.5 cm
    4. 15.5 cm

  85. Similar triangles areas 64 cm² and 49 cm². Altitude of first is 6cm. Altitude of second is

    1. 5.3
    2. 3.5
    3. 27.56
    4. 8.76

  86. ABC and DEF similar, BC=3.5cm, EF=2.5cm, area ABC=7 sq cm. Area DEF is

    1. 4.59
    2. 5.49
    3. 9.54
    4. 3.57

  87. Boy 90cm tall walks from base of 3.6m building at 1.2m/s. Shadow after 4 seconds is

    1. 1.6 m
    2. 1.7 m
    3. 1.8 m
    4. 2 m

  88. Two angles of triangle are congruent; third is 30°. Measure of one congruent angle is

    1. 45°
    2. 60°
    3. 75°
    4. 90°

  89. DE||BC, AE=AC/4, AB=6cm. Value of AD is

    1. 5 cm
    2. 1 cm
    3. 1.5 cm
    4. 2 cm

  90. In figure CAB~CED with given measurements, value of x is

    1. 3 m
    2. 6 m
    3. 9 m
    4. 12 cm

  91. Similar triangles have perimeters 25 cm and 15 cm. One side of first is 9 cm. Corresponding side of other is

    1. 1.4 cm
    2. 1.6 cm
    3. 3.4 cm
    4. 5.4 cm

  92. Student 90cm tall walks from lamp (3.6m high) at 1.2m/s. Shadow length after 4 seconds is

    1. 1 m
    2. 1.6 m
    3. 3.6 m
    4. 4.8 m

  93. Areas of similar triangles are 169 cm² and 121 cm². Longest side of larger is 26 cm. Longest side of smaller is

    1. 16 cm
    2. 22 cm
    3. 26 cm
    4. 42 cm

  94. Areas of two similar triangles are 81 cm² and 49 cm². Ratio of corresponding heights is

    1. 3:7
    2. 5:7
    3. 7:9
    4. 9:7

  95. 25m ladder reaches window 20m above ground. Distance of foot from building is

    1. 5 m
    2. 10 m
    3. 15 m
    4. 20 m

  96. Hypotenuse of right triangle is 6m more than twice shortest side. Third side is 2m less than hypotenuse. Sides are

    1. 10, 24 and 26
    2. 20, 24 and 26
    3. 20, 25 and 26
    4. 10, 25 and 26

  97. In figure showing a right triangle with altitude to hypotenuse, the ratio involving the segments is

    1. AB²/AD
    2. AD²/AB
    3. AB/AD²
    4. AD/AB²

  98. In triangle DEF, DE=6√3 cm, DF=12 cm, EF=6 cm. Angle E is

    1. 120°
    2. 90°
    3. 60°
    4. 45°

  99. In figure, ST||QR. If PS=?, given values from figure

    1. 2 cm
    2. 4.5 cm
    3. 4 cm
    4. 3 cm

  100. Trapezium ABCD, AB||DC, diagonals meet at O, AB=2CD. Ratio of areas of triangles AOB and COD is

    1. 1:2
    2. 2:1
    3. 1:4
    4. 4:1

  101. If one pair of corresponding sides are proportional and included angles equal, then the criterion is called

    1. AAA similarity
    2. SSS similarity
    3. SAS similarity
    4. ASA similarity

  102. P, Q on AB, AC of triangle ABC; PQ||BC divides ABC into 2 equal areas. AP:AB =

    1. 1:1
    2. 1:√2
    3. 1:√2
    4. (√2–1):1

  103. Using similar triangles in a figure, determine x

    1. (5√11)/4
    2. (3√11)/4
    3. (5√11)/3
    4. (3√11)/2

  104. Equilateral triangle ABC each side 4. AD⊥AB, AD=1. Area of BCED is

    1. 8
    2. 4
    3. 4.5
    4. 3.5

  105. Area of right triangle is 40 cm², perimeter 40 cm. Length of hypotenuse is

    1. 16 cm
    2. 18 cm
    3. 17 cm
    4. 19 cm

  106. x is length of median of equilateral triangle. Area is

    1. x²
    2. (x²/3)√3
    3. (x²√3)/3
    4. 2x²/3

  107. Two sides of triangle produced beyond base, exterior angles bisected; bisectors include angle equal to

    1. half sum of base angles
    2. sum of base angles
    3. half difference of base angles
    4. difference of base angles

  108. PSR right-angled at S, D mid-point of SR. Bisector of angle PSR and perp bisector of SR meet at O. Triangle OSD is

    1. scalene
    2. equilateral
    3. isosceles right angled
    4. acute-angled

  109. Equilateral triangle area X and square area Y have same perimeter. Then

    1. X>Y
    2. X<Y
    3. X=Y
    4. X≥Y

  110. In an isosceles triangle 'a' is the base and 'b' is the equal side. Its area is

    1. (a/4)√(4b²–a²)
    2. (b/4)√(4b²–a²)
    3. (a/4)√(4a²–b²)
    4. (b/4)√(4a²–b²)