CBSE Class 9 Areas of Parallelogram and Triangle Sure Shot Questions

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Study Material for Class 9 Mathematics Chapter 9 Areas of Parallelograms and Triangles

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Class 9 Mathematics Chapter 9 Areas of Parallelograms and Triangles

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1. ABCD is a parallelogram x and y are midpoints of BC and CD respectively. Prove that ar(ΔAXY) = 3/8(||^gmABCD)

2. The medians BE and CF of a triangle ABC intersect at G. Prove that ar(ΔGBC) = area of quadrilateral AFGE.

3. In fig. PQRS is a square and T and U are respectively, the midpoints of PS and QR. Find the area of ΔOTS if PQ = 8cm. useful-resources-areas-parallelogram-and-triangle-cbse-class-9

4. In fig. ABCD, ABFE and CDEF are parallelograms. Prove that ar(ΔADE) = ar(ΔBCF).

useful-resources-areas-parallelogram-and-triangle-cbse-class-9-2

5. In fig. ABCD is a trapezium in which AB || CD. Prove that area of ΔAOD = area of ΔBOC.

useful-resources-areas-parallelogram-and-triangle-cbse-class-9-3

6. The diagonals of parallelogram ABCD intersect at a point O. Through O, a line is drawn to intersect AD at P and BC at Q. Show PQ divides the parallelogram into two parts of equal area.

7. In the fig. O is any point on the diagonal BD of the parallelogram ABCD. Prove that ar(ΔOAB) = ar(ΔOBC).

CBSE Class 9 Areas of Parallelogram and Triangle Sure Shot Questions

8. Show that the diagonals of a parallelogram divide it into four triangles of equal area.

9. In fig. ABCD is a parallelogram and BC is produced to a point Q such that AD = CQ. If AQ intersects DC at P, show that ar(ΔBPC) = ar(ΔDPQ)

10. Prove that “Two parallelograms on the same base and between the same parallels are equal in area”.

11. Prove that “Two triangles on the same base and between the same parallels are equal in area”.

12. Prove that a median of a triangle divides it into two equal parts

13. If a triangle and a parallelogram are on the same base and between the same parallels, then prove that the area of the triangle is equal to half the area of the parallelogram.

14. If E,F,G and H are respectively the mid-points of the sides of a parallelogram ABCD, show that ar (EFGH) = 1/2 ar (ABCD)

15. Show that the diagonals of a parallelogram divide it into four triangles of equal area.

16. D and E are points on sides AB and AC respectively of ΔABC such that ar (DBC) = ar (EBC). Prove that DE || BC.

17. XY is a line parallel to side BC of a triangle ABC. If BE || AC and CF || AB meet XY at E and F respectively, show that ar (ABE) = ar (ACF)

18. In the below figure PSDA is a parallelogram. Points Q and R are taken on PS such that PQ = QR = RS and PA || QB || RC. Prove that ar (PQE) = ar (CFD).

19. X and Y are points on the side LN of the triangle LMN such that LX = XY = YN. Through X, a line is drawn parallel to LM to meet MN at Z (See Fig. 9.12). Prove that ar (LZY) = ar (MZYX)

20. The area of the parallelogram ABCD is 90 cm2. Find (i) ar (ABEF) (ii) ar (ABD) (iii) ar (BEF).

21. In ΔABC, D is the mid-point of AB and P is any point on BC. If CQ || PD meets AB in Q, then prove that ar (BPQ) = 1/2 ar (ABC).

22. ABCD is a square. E and F are respectively the midpoints of BC and CD. If R is the mid-point of EF, prove that ar (AER) = ar (AFR)

23. O is any point on the diagonal PR of a parallelogram PQRS. Prove that ar (PSO) = ar (PQO).

24. ABCD is a parallelogram in which BC is produced to E such that CE = BC. AE intersects CD at F. If ar (DFB) = 3 cm², find the area of the parallelogram ABCD.

25. In trapezium ABCD, AB || DC and L is the mid-point of BC. Through L, a line PQ || AD has been drawn which meets AB in P and DC produced in Q. Prove that ar (ABCD) = ar (APQD).

26. If the mid-points of the sides of a quadrilateral are joined in order, prove that the area of the parallelogram so formed will be half of the area of the given quadrilateral.

27. In the below figure, l, m, n, are straight lines such that l || m and n intersects l at P and m at Q. ABCD is a quadrilateral such that its vertex A is on l. The vertices C and D are on m and AD || n. Show that ar (ABCQ) = ar (ABCDP)

28. In the below figure, BD || CA E is mid-point of CA and BD = 1/2 CA. Prove that ar(ABC) = 2ar(DBC)

29. In the below figure, ABCD is a parallelogram. Points P and Q on BC trisects BC in three equal parts. Prove that ar (APQ) = ar (DPQ) = 1/6 ar(ABCD)

30. A point E is taken on the side BC of a parallelogram ABCD. AE and DC are produced to meet at F. Prove that ar (ADF) = ar (ABFC)

31. The diagonals of a parallelogram ABCD intersect at a point O. Through O, a line is drawn to intersect AD at P and BC at Q. Show that PQ divides the parallelogram into two parts of equal area.

32. In the below figure, X and Y are the mid-points of AC and AB respectively, QP || BC and CYQ and BXP are straight lines. Prove that ar (ABP) = ar (ACQ).

33. Parallelogram ABCD and rectangle ABEF are on the same base AB and have equal areas. Show that the perimeter of the parallelogram is greater than that of the rectangle.

34. In the below figure, ABCD and AEFD are two parallelograms. Prove that ar (PEA) = ar (QFD)

35. In the below figure, ABCDE is any pentagon. BP drawn parallel to AC meets DC produced at P and EQ drawn parallel to AD meets CD produced at Q. Prove that ar (ABCDE) = ar (APQ)

36. In the below figure, CD || AE and CY || BA. Prove that ar (CBX) = ar (AXY)

37. In fig. P is a point in the interior of a parallelogram ABCD. Show that

(i) ar (APB) + ar (PCD) = 1/2 ar (ABCD)

(ii) ar (APD) + ar (PBC) = ar (APB) + ar (PCD)

38. In Fig. ABCD is a quadrilateral and BE || AC and also BE meets DC produced at E. Show that area of ΔADE is equal to the area of the quadrilateral ABCD.

39. Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at O. Prove that ar (AOD) = ar (BOC).

40. Diagonals AC and BD of a quadrilateral ABCD intersect each other at O such that ar (AOD) = ar (BOC). Prove that ABCD is a trapezium.

41. ABCD is a trapezium with AB || DC. A line parallel to AC intersects AB at X and BC at Y. Prove that ar (ADX) = ar (ACY).

42. In the above Fig. AP || BQ || CR. Prove that ar (AQC) = ar (PBR).

43. Diagonals AC and BD of a quadrilateral ABCD intersect at O in such a way that ar (AOD) = ar (BOC). Prove that ABCD is a trapezium.

44. The medians BE and CF of a triangle ABC intersect at G. Prove that the area of ΔGBC = area of the quadrilateral AFGE.

45. Diagonals AC and BD of a quadrilateral ABCD intersect each other at P. Show that ar (APB) × ar (CPD) = ar (APD) × ar (BPC).

46. ABCD is a trapezium in which AB || DC, DC = 30 cm and AB = 50 cm. If X and Y are, respectively the mid-points of AD and BC, prove that ar (DCYX) = 7/9 ar (XYBA)

47. In ΔABC, if L and M are the points on AB and AC, respectively such that LM || BC. Prove that ar (LOB) = ar (MOC)

48. If the medians of a ΔABC intersect at G, show that ar (AGB) = ar (AGC) = ar (BGC) = 1/3 ar (ABC)

49. Prove that the area of rhombus is equal to half the rectangle contained by its diagonals.

50. A point O inside a rectangle ABCD is joined to the vertices. Prove that the sum of the areas of a pair of opposite triangles so formed is equal to the sum of the other pair of triangles.

51. The medians BE and CF of a triangle ABC intersect at G. Prove that area of ΔGBC = area of quadrilateral AFGE.

52. A villager Itwaari has a plot of land of the shape of a quadrilateral. The Gram Panchayat of the village decided to take over some portion of his plot from one of the corners to construct a Health Centre. Itwaari agrees to the above proposal with the condition that he should be given equal amount of land in lieu of his land adjoining his plot so as to form a triangular plot. Explain how this proposal will be implemented.

53. P and Q are respectively the mid-points of sides AB and BC of a triangle ABC and R is the midpoint of AP, show that

(i) ar (PRQ) = 1/2 ar (ARC) (ii) ar (RQC) = 3/8 ar (ABC) (iii) ar (PBQ) = ar (ARC)

54. A quadrilateral ABCD is such that the diagonal BD divides its area in two equal parts. Prove that BD bisects AC.

55. D, E and F are respectively the mid-points of the sides BC, CA and AB of a ABC. Show that

(i) BDEF is a parallelogram. (ii) ar (DEF) = 1/4ar (ABC) (iii) ar (BDEF) = 1/2 ar (ABC).

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CBSE Class 9 Mathematics Chapter 9 Areas of Parallelograms and Triangles Study Material

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