MATH ASSIGNMENT CLASS -IX AREAS OF PARALLELOGRAM AND TRIANGLES
1. ABCD is a rectangle. P,Q,R and S are the mid-points of AB,BC,CD and DA respectively. Prove that triangles APS, BPQ, CQR and DRS have equal area. 2. Show that a median of a triangle divides it into two triangles of equal area. 3. A point D is taken on the side BC of a triangle ABC such that BD = 2DC. Prove that ar (∆ABD) = 2 ar (∆ADC). 4. If a parallelogram and a square (or a rectangle) stand on the same base and between the same parallels, then the perimeter of the square (or the rectangle) is less than that of the parallelogram.
5. In the given fig1 ABCD is a parallelogram. Prove that ar (∆BCP) = ar (∆DPQ).
6. If the ratio of the altitude and the area of parallelogram is 2 : 11, Then find the length of the base of parallelogram. 7. Prove that area of trapezium = 8.
1 2
(sum of parallel sides) h
In the fig2, ABCD is a parallelogram in which BC is produced to E such that CE = BC. AE intersects CD at F. Show that ar (ΔBDF) =
9.
1 4
Prove that area of equilateral triangle of side a units is
ar (ABCD)
3 a
4
2
sq. units.
10. In the given figure 3, ABC is a triangle. D is mid-point of AB, P is any point on BC. Line CQ is drawn parallel to PD intersect AB at Q. PQ is joined. Show that area
BPQ =
1 2
area ABC
A
Q D
fig3 B
P
C
11. In the figure4, ABCD and AEFD are two parallelograms. Prove that ar (PEA) = ar (QFD).
12. In the fig6, ABCD is a parallelogram. E and F are mid points of BC and DC respectively.
Prove that ar (∆AEF) =
3 8
ar( ABCD).
13. In the fig6, fig6, ABCD is a square. E and F are respectively respectively the mid-points mid-points of BC and CD. If If R is the mid point of EF, show that ar (AER) = ar (AFR).
14. Median AD and CE of ABC ABC intersect at M. The mid point of AE is N. If the area of MNE is k times the area of ABC ABC , find the value of k , MNE A
N E M
B
C
15. EFGH is a parallelogram. A is a point in EH. B is the foot of the perpendicular from G to AF. Prove that ar ( EFGH ) AF GB
.
16. 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. A
D
P
O
B
C
Q
17. Through the vertex A of a parallelogram ABCD, line AEF is drawn to mee t BC at E and DC produced at F. Show that the triangles BEF and DCE are equal in area.
D
C
F
E
A
B
18. ABCD is a trapezium in which AB ∥ DC and DC = 40 cm and AB = 60 cm. If X and Y are respectively, the mid points of AD and BC, prove that 9 (i) XY = = 50 cm (ii) DCYX is a trapezium (iii) ar ( DCYX ) ar ( XYBA) 11
19. ABCD is a quadrilateral. A line through D, parallel to AC, meets BC produced in P. prove that that ar (∆ABP) = ar (quad (quad ABCD).
20. PQRS is a parallelogram whose area is 180 cm² and A is any point on the diagonal QS. The area of Δ ASR = 90cm².
