CLASS X: TRIANGLES
(PREVIOUS YEARS BOARD QUESTION PAPERS)
1. In A ABC, DEIIBC. IF DE = ~ BC and area of A ABC = 81 cm 2, find the
area of A ADE. (CBSE 2009-1M)
2. In ALMN, LL= 60°, LM= 50°. If ALMN-APQR. findLR.
(CBSE 2010-1M)
3. In fig 1, PT=2 cm, TR=4 cm, STIIQR. Find the ratios of are of A PST
& A PQR. (CBSE 2010-1M)
4. In fig.2, A AHK-A ABC. If AK=10Cm, BC=3.5 cm, HK=7 cm. Find
AC. (CBSE 2010-1M)
5. In fig 5, MNIIAB, BC=7 .5cm, AM=4Cm, MC=2Cm. Find BN.
(CBSE2010-1 M)
6. In A PRQ, ST II RQ, PS = 3 cm and SR = 4 cm. Find the ratio of the
area of A PST to the area of A PRQ. (CBSE2017-1M)
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ar /J.ABC
7. GIven
A
-A
, I PQ = , then find ar ll.PQR. (CBSE2018- '·l,l
3
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1M)
8. In fig-2. ABIIDE, BDIIEF. Prove that DC2 =CFxAC. L R .(CBSE 2007, 2010-2M)
9. In A ABC, AB=AC, D is a point on side AC such that c
2
BC =ACxCD. Prove that BD=BC. (CBSE 2010-2M)
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10. P and Q are points on sides CA and CB respectively
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of ABC, right angled at c. Prove that
AQ2+BP2=AB2 +PQ2• (CBSE 2007-2M)
11. In. Fig. 9, PQ = 24 cm, QR = 26 cm, L PAR = 90°,
! Cu..a.....=..;=.:...~B
PA = 6 cm and AR = 8 cm. Find L QPR. (CBSE
Flgure-4
Figure - 2
p
2008-2M)
Q...,....-- - - -=
12. P and Q are points on the sides AB and AC respectively of D. ABC
such that AP =3.5 cm, PB = 7 cm, AQ = 3 cm and QC = 6 cm. If
PQ = 4.5 cm. find BC. (CBSE 2008-2M)
13. The perpendicular AD on the base BC of 6. ABC
:'.
intersects BC in D such that BD=3CD. Prove that
,,
2AB2=2AC 2+BC2 . (CBSE 2010-lM)
14. In fig4, ABC is a right angles triangle, right angled at
C.
DE l. AB. Prove that A ABC- A ADE. hence find the
lengths of AE and DE. (CBSE 2010-lM)
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15. In fig.3, ABC is right angled at c . D is the midpoint of j1'
BC. Prove that AB 2=4AD2-3AC 2 . (CBSE 2010-lM)
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16. In A ABC, right angles at A, BL and CM are two t
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medians. Prove that 4(BL 2+CM2) =5BC 2.
... ~ : ....
(CBSE 2010-lM)
17. In given figure 7, L1 = L2 & A NSQ A MTR then prove that A
PTS - A PRQ. (CBSE 2017-lM)
18. Prove that the area of an equilateral triangle described on one
side of the square is equal to half the area of the equilateral M
Q
N
R
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t
lo
2ck
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~
r
1
=
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·. ...-=--;;...:.:-+--~ K-
triangle described on one of its diagonal. (CBSE 2018-3M)
19. If the diagonals of a quadrilateral divide each other proportionally prove that it is a
trapezium. (CBSE 2008-3M)
20. Two triangles ABC and DBC are on the same base BC and on the same side of BC
in which LA= L D = 90°. If CA and BD meet each other at E, show that AE.EC =
BE.ED. (CBSE 2008-3M)
21. In an equilateral triangle ABC(Fig 8), D is a point on the side BC such that BO=;
BC. Prove that 9AD2 =7AB2 • (CBSE 2017-3M & 2018-4M)
22. If the area of two similar triangles are equal, prove that they are congruent. (CBSE
2018-3M)
23. The perpendicular from A on side BC of a a.ABC meets BC at D such that DB=3CD.
Prove that 2AB2 =2AC 2 +BC2 • CBSE2019-lM
24.AD and PM are medians of triangles ABC and PQR respectively where
AB
AD
MBC ~~PQR. Prove that PQ = PM • CBSE2019 - 3M
25. Prove that the ratio of the areas of two similar triangles is equal to the
ratio of the squares of their corresponding sides. Using the above
theorem prove the following: The area of the equilateral triangle
described on the side of a square is half the area of the equilateral
triangle described on its diagonal. (CBSE 2008, 2009, 2010-4M)
26. Prove that the ratio of the areas of two similar triangles is equal to the
ratio of the squares of their corresponding sides. Also, in fl ABC, XY is
parallel to BC and it divides triangle ABC into two parts of equal area.
A
8
D
Prove that BX = ✓
5i . (CBSE 2008-&M)
AB
2
27. Prove that in a triangle. if square of one side is equal to the sum of the squares of
the other two sides, then angle opposite to first side is right angle. (CBSE 2010-4M)
28. Prove that if the areas of two similar triangles are equal then the
R
triangles are congruent. (CBSE 2010-4M)
29. If a line is drawn parallel to one side of a triangle to intersect other
two sides in distinct points, prove that the other two sides are divided
in the same ratio. Usin~ the above. do the following. In fig 6, PQIIAB,
AQIICB. Prove that AR =PRxCR. (CBSE 2007, 2010-4M)
30. If a line is drawn parallel to one side of a triangle to intersect other
two sides in distinct points, prove that the other two sides are divided
in the same ratio. Also, In triangle ABC, DEIIBC and BD=CE. Prove P
that ABC is an isosceles triangle. (CBSE 2007, 6M)
31. Prove that, in a right triangle, the square on the hypotenuse is equal to
the sum of the squares on the other two sides. (CBSE 2018-4M)
32. Two triangles ABC and DBC are on the same base BC in which L A =
L D = 90°. If CA and BO meet each other at E, show that AE X CE =BEX B
•
p
ED. (CBSE SQP-2M)
33. The diagonals of a trapezium ABCD with ABIIDC intersect each other at
point o. If AB = 2CD, find the ratio of the areas of triangles AOB and coo.
C
1
Q
~
(CBSE SQP-3M)
34. In figure, sand trisect the side QR of a right triangle PQR.
Prove that: 8PT2 =3PR2+5PS2 (CBSE SQP-3M)
35. In figure, POIIAB and PRIIAC. Prove that QRIIBC. (CBSE
SQP-2M)
B
C
C
36. In figure, PA, QB and RC are each perpendicular to AC. Prove
A
~
,.
that ]_ + ]_ = ]_ .(CBSE SQP-4M)
X
J
Z
8
37.
In an isosceles triangle ABC with AB = AC, BD
A
2
2
is perpendicular from B to the side AC. Prove that BD -CD = 2CDxAD.
(CBSE SQP-4M)
38.
If a line is drawn parallel to one side of a triangle to intersect other
two sides in distinct points, prove that the other two sides are divided in the same
ratio. (CBSE 2019-4M)
39. Prove that in a right triangle, the square on the hypotenuse is equal to the sum of
the squares on the other two sides. (CBSE 2019-4M)