# ABCD is a parallelogram. its perimeter is 20 cm. if (^ABD)=60, BD=8 CM, calculate AB, AD.

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We are given a parallelogram ABCD with a perimeter of 20 cm. The side BD = 8 cm and angle ABD= 60 degrees.

As ABCD is a parallelogram AB = CD and AD = BC

The perimeter of the parallelogram is AB + BC + CD + DA = 20 cm

=> AB + AD = 20/2 = 10 cm

If AB = x, AD = 10 - x

BD = 8 cm

Apply the cosine rule to triangle ABD

AB^2 + BD^2 - 2*AB*BD*cos ABD = AD^2

x^2 + 8^2 - 2*x*8*cos 60 = (10 - x)^2

=> x^2 + 64 - 8x = 100 + x^2 - 20x

=> 12x = 36

=> x = 36/12 = 3

**This gives AB = 3 cm and AD = 7 cm.**

ABCD is a parallelogram.

Also given the perimeter AB+BC+CD+AD = 20 cm

Therefore AB+AD = 1/2 (20 cm) = 10 cm, as opposite sides of a parallelogram are equal in length.

Let AB = X. Then AD = 10-x. Given BD = 8 cm, angle ABD = 60 deg.

We apply cosine rule for triangle ABD:

AB^2+BD^2- 2AB*BD = AD^2.

x^2+8^2-2*x*8*cos60 = (10-x)^2

x^2+64-2*x*8*(1/2) = (10-x)^2

x^2+64-8x = 10^2-20x+x^2.

=> 64-8x = 100-20x.

=> 20x - 8x = 100-64 = 36

=> 12x = 36

=> x = 36/12 = 3. Or AB = 3 cm. So AM = (10-x)cm = (10 -3) cm = 7 cm.

**Therefore AB = 3cm. AD = 7 cm**.