Question

ABCD is a parallelogram in which AE is perpendicular to CD as 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.

Answer 1

Given area of ΔAED = 6 cm² and AC = 5 cm and DE = 4 cm

∴ Area of ΔAED = `1/2` × DE × AE  ......[∵ Area of triangle = Base × Height]

⇒ `1/2` × 4 × AE = 6

⇒ AE = `(6 xx 2)/4`

⇒ AE = 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 = 5² – 3² = 25 – 9

⇒ EC = `sqrt(16)`

⇒ EC = 4 cm

∵ DE + EC = DC

⇒ DC = 4 + 4 = 8 cm

∵ ABCD is a parallelogram

So, AB = DC = 8 cm

Now, In right-angled ΔAED, AD² = AE² + ED²  ......[By Pythagoras theorem]

⇒ AD² = 3² + 4² = 9 + 16

⇒ AD = `sqrt(25)`

⇒ AD = 5 cm

So, AD = BC = 5 cm  ......[∵ ABCD is a parallelogram]

∴  Perimeter of paralielogram 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 cm.