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Question 102

ABCD is a parallelogram in which AE is perpendicular to CD is shown in the given figure. Also, AC = 5 cm, DE = 4 cm and area of ΔAED=6 cm2.
Find the perimeter and area of parallelogram ABCD.

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Solution

Given, area ΔAED=6 cm2 and AC = 5 cm and DE = 4 cm
Area of ΔAED=12×DE×AE [ area of triangle =12×base×height]
12×4×AE=6AE=6×24AE=3 cm
Now, in right angled ΔAEC, AE = 3 cm and AC = 5 cm
So, (EC)2=(AC)2(AE)2 [by Pythagoras theorem]
(EC)2=5232=259EC=16EC=4 cmDE+EC=DCDE+EC=DCDC=4+4=8 cm
ABCD is a parallelgoram.
So, AB = DC = 8 cm
Now, in right angled ΔADE,AD2=AE2+ED2
AD2=32+42=9+16AD=25AD=5 cm
So, AD = BC = 5 cm [ ABCD is a parallelogram]
Perimeter of paralelogram ABCD =2(l+b)=2(DC+AD)=2(8+5)=2×13=26 cm
Area of parallelogram ABCD =Base×Height=DC×AE=8×3=24 cm2


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