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Triangles Class 10 Maths CBSE Important Questions With Solutions

Important Questions for Class 10 Maths Chapter 6 Triangles with solutions includes all the important topics with detailed explanation that aims to help students to score more marks in Board Exams 2020. Students who are preparing for their Class 10 exams must go through Important Questions for Class 10 Math Chapter 6 Triangles.

Important Questions for Class 10 Maths Chapter 6 Triangles

Expert teachers at CBSETuts.com collected and solved 2 Marks and 4 mark important questions for Class 10 Maths Chapter 6 Triangles.

MathsScienceEnglishSocialHindi AHindi B

2016

Very Short Answer Type Question [1 Mark]

Question 1.
If A ABC ~ ARPQ, AB = 3 cm, BC = 5 cm, AC = 6 cm, RP = 6 cm and PQ = 10 cm, then find QR
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 1

Short Answer Type Question I [2 Marks]

Question 2.
R and S are points on the sides DE and EF respectively of a ADEF such that ER = 5 cm, RD = 2.5 cm, SE = 1.5 cm and FS = 3.5 cm. Find whether RS || DF or not.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 2

Short Answer Type Questions II [3 Marks]

Question 3.
From airport two aeroplanes start at the same time. If the speed of first aeroplane due North is 500 km/h and that of other due East is 650 km/h, then find the distance between two aeroplanes after 2 hours.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 3

Question 4.
AABC, is right angled at C. If p is the length of the perpendicular from C to AB and a, b, c are the lengths of the sides opposite ∠A,∠B, ∠C respectively then prove that Triangles Class 10 Maths CBSE Important Questions With Solutions 4
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 5
Triangles Class 10 Maths CBSE Important Questions With Solutions 6

Question 5.
In the figure, ABCD is a parallelogram and E divides BC in the ratio 1: 3. DB and AE intersect at F. Show that DF = 4FB and AF = 4FE
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 7

Long Answer Type Questions [4 Marks]

Question 6.
State and prove Converse of Pythagoras’ Theorem.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 8

Question 7.
In the figure, PQR and QST respectively. Prove that QR X QS= QP X QT
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 9
Triangles Class 10 Maths CBSE Important Questions With Solutions 10

2015

Very Short Answer Type Question [1 Mark]

Question 8.
In ADEW, AB || EW. If AD = 4 cm, DE = 12 cm and DW = 24 cm, then find the value of DB.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 11
Triangles Class 10 Maths CBSE Important Questions With Solutions 12

Short Answer Type Question I [2 Marks]

Question 9.
A ladder is placed against a wall such that its foot is at distance of 5 m from the wall and its top reaches a window 5/3 m above the ground. Find the length of the ladder
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 13

Short Answer Type Questions II [3 Marks]

Question 10.
In figure, if ∠CAB = ∠CED, then prove that AB X DC = ED X BC.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 14
Triangles Class 10 Maths CBSE Important Questions With Solutions 15

Question 11.
State whether the given pairs of triangles are similar or not. In mention the criterion.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 16
Triangles Class 10 Maths CBSE Important Questions With Solutions 17

Long Answer Type Questions [4 Marks]

Question 12.
In ∆ABC, from A and B altitudes AD and BE are drawn. Prove that ∆ADC ~ ∆BEC. Is ∆ADB ~ ∆AEB and ∆ADB ~ ∆ADC?
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 18

Question 13.
In ∆ABC, if ∠ADE = ∠B, then prove that ∆ADE ~ ∆ABC. Also, if AD = 7.6 cm, AE = 7.2 cm, BE = 4.2 cm and BC = 8.4 cm, then find DE.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 19

2014

Very Short Answer Type Questions [1 Mark]

Question 14.
If ∆ABC ~ ∆RPQ, AB = 3 cm, BC = 5 cm, AC = 6 cm, RP = 6 cm and PQ = 10 cm, then find QR.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 20

Question 15.
Let  ∆ABC ~ ∆DEF, ar (∆ABC) = 169 cm2 and ar (∆DEF) = 121 cm2. If AB = 26 cm2 then find DE.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 21

Question 16.
If in an equilateral triangle, the length of the median is √3 cm, then find the length of the side of equilateral triangle.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 22
Triangles Class 10 Maths CBSE Important Questions With Solutions 133

Short Answer Type Questions I [2 Marks]

Question 17.
In the figure, D andE are points on AB and AC respectively such that DE||BC. If AD = 1/3 BD and AE = 4.5 cm, find AC.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 23
Triangles Class 10 Maths CBSE Important Questions With Solutions 24

Question 18.
Determine whether the triangle having sides (a – 1) cm, 2 √a cm and (a + 1) cm is a right angled triangle.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 25

Question 19.
In an equilateral triangle of side 3√3 cm, find the length of the altitude.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 26
Triangles Class 10 Maths CBSE Important Questions With Solutions 27

Short Answer Type Questions II [3 Marks]

Question 20.
In the figure, ABC is a triangle and BD ⊥ AC. Prove that AB2 + CD2 = AD2 + BC2
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 28

Question 21.
Right angled triangles BAC and BDC are right angled at A and D and they are on same side of BC. If AC and BD intersect at P, then prove that AP x PC = PB X DP.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 29

Question 22.
In the given figure, if LM || CB and LN|| CD, prove that AM X AD = AB X AN.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 30
Triangles Class 10 Maths CBSE Important Questions With Solutions 31

Question 23.
In the given figure, ABC is a triangle, right angled at B and BD⊥AC. If AD = 4 cm and CD = 5 cm, find BD and AB.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 32

Question 24.
Equiangular triangles are drawn on sides of right angled triangle in which perpendicular is double of its base. Show that area of triangle on the hypotenuse is the sum of areas of the other two triangles?
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 33

Question 25.
If in a right angle AABC, right angled at A, AD ⊥ BC, then prove that AB2 + CD2 = BD2 + AC2
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 34

Long Answer Type Questions [4 Marks]

Question 26.
Prove that the ratio of the areas of two similar triangles is equal to the ratio of squares of their corresponding sides.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 35

Question 27.
In ∆ABC, AX⊥ BC and Y is middle point of BC. Then prove that,

Triangles Class 10 Maths CBSE Important Questions With Solutions 36
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 37
Triangles Class 10 Maths CBSE Important Questions With Solutions 38

Question 28.
Prove that if a line is drawn parallel to one side of a triangle to intersect the othertwo sides at distinct points, then other two sides are divided in the same ratio.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 39
Triangles Class 10 Maths CBSE Important Questions With Solutions 40

Question 29.
Prove that the sum of square of the sides of a rhombus is equal to the sum of squares of its diagonals.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 41

Question 30.
In AABC, X is any point on AC. If Y, Z , U and Y are the middle points of AX, XC, AB and BC respectively, then prove that UY || VZ and UV|| YZ.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 42
Triangles Class 10 Maths CBSE Important Questions With Solutions 43

Question 31.
In ∆ABC, ZB = 90°, BD ⊥ AC, ar (∆ABC) = A and BC = a, then prove that
Triangles Class 10 Maths CBSE Important Questions With Solutions 44
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 45
Triangles Class 10 Maths CBSE Important Questions With Solutions 46
Triangles Class 10 Maths CBSE Important Questions With Solutions 47

2013

Short Answer Type Questions I [2 Marks]

Question 32.
In the figure, PQR and SQR are two right triangles with common hypotenuse QR. If PR and SQ intersect at M such that PM = 3 cm, MR = 6 cm and SM = 4 cm, find the length of MQ.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 48
Triangles Class 10 Maths CBSE Important Questions With Solutions 49

Question 33.
Find the length of each altitude of an equilateral triangle
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 50
Triangles Class 10 Maths CBSE Important Questions With Solutions 51

Short Answer Type Questions Il [3 Marks]

Question 34.
In the figure, DB⊥BC, DE⊥AB and AC⊥ BC prove that
Triangles Class 10 Maths CBSE Important Questions With Solutions 52
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 53
Triangles Class 10 Maths CBSE Important Questions With Solutions 54

Question 35.
In the given figure, ABCD is a rectangle. P is the mid-point of DC. If QB = 7 m, AD = 9 cm and DC = 24 cm, then prove that ∠APQ = 90°
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 55
Triangles Class 10 Maths CBSE Important Questions With Solutions 56
Triangles Class 10 Maths CBSE Important Questions With Solutions 57

Question 36.
In AABC, D, E and F are the mid-points of AB, BC and AC respectively. Find the ratio of area of ∆ABC and area of ∆DEF.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 58

Long Answer Type Questions [4 Marks]

Question 37.
In the figure, ABC is a right triangle, right angled at B. AD and CF are two medians drawn from A and C respectively. If AC = 5 cm and AD = 3√5/2cm. Find the length of CE.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 59

Question 38.
ABC is a right triangle, right-angled at B. D and E trisect BC, prove that 8AE2 = 3AC2 + 5 AD2
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 560
Triangles Class 10 Maths CBSE Important Questions With Solutions 60

Question 39.
Prove that in a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 61

Question 40.
In ∆ABC, AD is the median to BC and in ∆PQR, PM is the median to QR.
Triangles Class 10 Maths CBSE Important Questions With Solutions 62
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 63

2012

Short Answer Type Question I [2 Marks]

Question 41.
In the given figure, if AB || DC, find the value of x.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 64
Triangles Class 10 Maths CBSE Important Questions With Solutions 65

Short Answer Type Question II [3 Marks]

Question 42.
In the given figure PQ || BA; PR || CA. If PD = 12 cm. Find BD X CD.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 66
Triangles Class 10 Maths CBSE Important Questions With Solutions 67
Triangles Class 10 Maths CBSE Important Questions With Solutions 68

2011

Short Answer Type Question I [2 Marks]

Question 43.
If one diagonal of a trapezium divides the other diagonal in the ratio 1 : 3. Prove that one of the parallel sides is three times the other.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 69

Short Answer Type Questions II [3 Marks]

Question 44.
In given figure ∆ABC is similar to ∆XYZ and AD and XE are angle bisectors of ∠A and ∠X respectively such that AD and XE in centimetres are 4 and 3 respectively, find the ratio of area of ∆ABD and area of ∆XYE.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 70
Triangles Class 10 Maths CBSE Important Questions With Solutions 71
Triangles Class 10 Maths CBSE Important Questions With Solutions 72

Question 45
In figure, AB || PQ || CD, AB = x units, CD =y units and PQ = z units, prove that
Triangles Class 10 Maths CBSE Important Questions With Solutions 73
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 74
Triangles Class 10 Maths CBSE Important Questions With Solutions 75
Triangles Class 10 Maths CBSE Important Questions With Solutions 76

Long Answer Type Questions [4 Marks]

Question 46.
The area of two similar triangles are 49 cm2 and 64 cm2 respectively. If the difference of the corresponding altitudes is 10 cm, then find the lengths of altitudes (in centimetres).
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 77

Question 47.
In an equilateral triangle ABC, D is a point on side BC such that 4BD = BC. Prove that ADcm2 = BCcm2.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 78
Triangles Class 10 Maths CBSE Important Questions With Solutions 79

2010

Very Short Answer Type Questions [1 Mark]

Question 48.
In figure, S and T are points on the sides PQ and PR, respectively of APQR, such that PT = 2 cm, TR = 4 cm and ST is parallel to QR. Find the ratio of the areas of ∆PST and ∆PQR.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 80
Triangles Class 10 Maths CBSE Important Questions With Solutions 81

Question 49.
In figure, DE || BC in AABC such that that BC = 8 cm, AB = 6 cm and DA = 1.5 cm. Find DE.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 82
Triangles Class 10 Maths CBSE Important Questions With Solutions 83

Question 50.
In figure, MN || AB, BC = 7.5 cm, AM = 4 cm and MC = 2 cm. Find the length BN.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 84
Triangles Class 10 Maths CBSE Important Questions With Solutions 85 Triangles Class 10 Maths CBSE Important Questions With Solutions 86Triangles Class 10 Maths CBSE Important Questions With Solutions 86

Short Answer Type Questions I [2 Marks]

Question 51.
Triangle ABC is right angled at B, and D is mid-point of BC. Prove that AC2 = 4AD2 – 3AB2.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 87

Question 52.
If BL and CM are medians of a triangle ABC right angled at A, then prove that4(BL2 + CM2) = 5BC2.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 88
Triangles Class 10 Maths CBSE Important Questions With Solutions 89

Question 53.
In figure, a triangle ABC is drawn to circumscribe a circle of radius 3 cm, such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 6 cm and 8 cm respectively. Find the side AB if the area of ∆ABC = 63 cm2.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 90
Triangles Class 10 Maths CBSE Important Questions With Solutions 91

Short Answer Type Question II [3 Marks]

Question 54.
In figure, ABC is an isosceles triangle in which AB = AC. E is a point on the side CB produced, such that FE ⊥ AC. If AD ⊥ CB, prove that AB X EF = AD X EC.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 92
Triangles Class 10 Maths CBSE Important Questions With Solutions 93

Long Answer Type Questions [4 Marks]

Question 55.
Prove that in a right angle triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides. Point D is the mid-point of the side BC of a right triangle ABC, right angled at C. Prove that, 4AD2 = 4AC2 + BC2..
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 94

Question 56.
Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
Using the above, prove the following:
If the areas of two similar triangles are equal, then prove that the triangles are congruent.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 95

Question 57.
In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then prove that the angle opposite the first side is a right angle.
Using the above, do the following:
In an isosceles triangle PQR, PQ = QR and PR2 = 2PQ2. . Prove that ZQ is a right angle.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 96

Question 58.
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, prove that the other two sides are divided in the same ratio.
Using the above, do the following:
In figure, BA || QR, and CA || SR, prove
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 97
Triangles Class 10 Maths CBSE Important Questions With Solutions 98
Triangles Class 10 Maths CBSE Important Questions With Solutions 99

Question 59.
Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Using the above, do the following:
AD is an altitude of an equilateral triangle ABC. On AD as base, another equilateral triangle ADE is constructed. Prove that, area (∆ADE): area (∆ABC) =3:4
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 100
Triangles Class 10 Maths CBSE Important Questions With Solutions 101

Question 60.
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, prove that the other two sides are divided in the same ratio.
Using the above, do the following:
In figure, PQ || AB and AQ || CB.
Prove that AR2 = PR. CR.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 102
Triangles Class 10 Maths CBSE Important Questions With Solutions 103

2009

Very Short Answer Type Questions [1 Mark]

Question 61.
In fig. ∠M =∠N = 46°, express JC in terms of a, b andc, where a, andc are lengths ofLM, MN and NK respectively.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 104
Triangles Class 10 Maths CBSE Important Questions With Solutions 105

Question 62.
If the areas of two similar triangles are in ratio 25 : 64, write the ratio of their corresponding sides.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 106

Question 63.
In a ∆ABC, DE || BC. If DE = – BC and area of ∆ABC = 81 cm2, find the area of ∆ADE.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 107

Short Answer Type Questions I [2 Marks]

Question 64.
In figure,∆ABD is a right triangle, right angled at A and AC⊥BD. Prove that AB2 = BC.BD.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 108
Triangles Class 10 Maths CBSE Important Questions With Solutions 109

Question 65.
In figure, S and T trisect the side QR of a right triangle PQR.Prove that 8PT2 = 3PR2 + 5PS2.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 110
Triangles Class 10 Maths CBSE Important Questions With Solutions 111
Triangles Class 10 Maths CBSE Important Questions With Solutions 112

Short Answer Type Questions II [3 Marks]

Question 66.
In figure, AABC is right angled at C and DE ⊥ AB. Prove that ∆ABC ~ ∆ADE and hence find the lengths of AE and DE.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 113
Triangles Class 10 Maths CBSE Important Questions With Solutions 114

Question 67.
In figure, DEFG is a square and ∠BAC = 90°. Show that DE2 = BD X EC.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 115
Triangles Class 10 Maths CBSE Important Questions With Solutions 116
Triangles Class 10 Maths CBSE Important Questions With Solutions 117

Question 68.
In figure, AD ⊥ BC and BD = 1/3 CD.
Prove that 2CA2 = 2AB2 + BC2.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 118

Question 69.
In figure, M is mid-point of side CD of a parallelogram ABCD. The line BM is drawn intersecting AC at L and AD produced at E. Prove that EL = 2BL.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 119
Triangles Class 10 Maths CBSE Important Questions With Solutions 120

Question 70.
In figure, two triangles ABC and DBC lie on the same side of base BC. P is a point on BC such that PQ || BA and PR || BD. Prove that QR || AD.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 121
Triangles Class 10 Maths CBSE Important Questions With Solutions 122

Long Answer Type Questions [4 Marks]

Question 71.
Prove that the ratio of areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.
Using the above, do the following:
In a trapezium ABCD, AC and BD intersecting at O, AB|| DC and AB = 2CD, if area of ∆AOB = 84 cm2, find the area of ∆COD
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 123
Triangles Class 10 Maths CBSE Important Questions With Solutions 124

Question 72.
Prove that in a right angle triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides. ‘
Using the above, do the following:
Prove that in a ∆ABC, if AD is perpendicular to BC, then AB2 + CD2 = AC2 + BD2.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 125
Triangles Class 10 Maths CBSE Important Questions With Solutions 126
Triangles Class 10 Maths CBSE Important Questions With Solutions 127

Question 73.
Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided is the same ratio.
Using the above result, do the following:
In figure, DE|| BC and BD = CE. Prove that ∆ABC is an isosceles triangle.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 128
Triangles Class 10 Maths CBSE Important Questions With Solutions 129

Question 74.
Prove that the ratio of areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.
Using the above, prove the following: The areas of the equilateral triangle described on the side of a square is half the area of the equilateral triangle described, on its diagonal.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 130
Triangles Class 10 Maths CBSE Important Questions With Solutions 131

Question 75.
AABC is an isosceles triangle in which AC = BC. If AB2  =   2AC2 then, prove that ∆ABC is right triangle.
Solution:
Triangles Class 10 Maths CBSE Important Questions With Solutions 132

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