9th Area of Parallelogram and Triangle

CBSE ADDA
Area of Parallelogram and Triangle


Prove that followings:

1. Parallelograms on the same base and between the same parallels are equal in area. 
2. Two triangles on the same base (or equal base) and between the same parallels are equal in area. 
3. Two triangles having the same base (or equal bases) and equal areas lie between the same parallels. 
4. If a triangles 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. 
5. In ABCD is parallelogram and EFCD is a rectangle.
6. Also, AL ┴ DC. Prove that(i) ar (ABCD) = (EFCD)(ii) ar (ABCD) = DCxAL.
7. ABCD is a parallelogram, AE DC and CF AD. If AB =16cm, AE=8cm and CF=10cm, find AD. 
8. 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). 
9. P and Q are any two points lying on the sides DC and AD respectively of parallelogram ABCD. Show that ar (APB) =ar (BQC). 
10. P is a point in the interior of a parallelogram ABCD. Show that(i) ar (APB) +ar ( PCD) =!/2 ar ( ABCD)(ii) ar (APD) +ar (PBC) =ar (APB) + ar (PCD) 
11. In a triangle ABC, E is the mid- point of median AD. Show that ar (BED) =1/4 ar (ABC). 
12. Show that the diagonals of a parallelogram divide it into four triangle of equal area. 
13. D, E and F are respectively the mid- points of the sides BC, CA and AB of a ΔABC show that(i) BDEF is a parallelogram.(ii) ar (DEF)= ¼ ar (ABC)(iii) ar (BDEF)= ½ ar (ABC). 
14. D and E are points on sides AB and AC respectively of ΔABC such that ar (DBC) = ar (EBC). Prove that DE||BC. 
15. 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) 
16. Diagonals AC and BD of a trapezium ABCD with AB||DC intersect each other at O. Prove that ar (AOD) =ar (BOC). 
17. 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). 
18. Diagonal 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.