Question 1196783
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Part (a)


Diagram:
*[illustration parallelogram2.png]
Draw a segment from A to C. This is the longer diagonal.


Focus on triangle ABC
Side 'a' is opposite angle A
Side 'b' is opposite angle B
Side 'c' is opposite angle C


For triangle ABC, we have
a = 348
b = unknown
c = 424
angle B = 130 degrees


Apply the law of cosines.
Make sure your calculator is in degree mode.
b^2 = a^2+c^2 - 2*a*c*cos(B)
b^2 = 348^2+424^2 - 2*348*424*cos(130)
b^2 = 490,569.194768937
b = sqrt(490,569.194768937)
b = 700.406449691132
<font color=red>The longer diagonal (AC) is approximately 700.406 mm long</font>


Return to the original parallelogram diagram shown at the top. Draw a line from B to D to form triangle ABD


Angle DAB is 180-(angleABC) = 180-130 = 50 degrees
This is angle A of triangle ABD


We have this info about triangle ABD
side b = 348
side d = 424
angle A = 50 degrees


Use the law of cosines
a^2 = b^2+d^2 - 2*b*d*cos(A)
a^2 = 348^2+424^2 - 2*348*424*cos(50)
a^2 = 111,190.805231063
a = sqrt(111,190.805231063)
a = 333.452853085802
<font color=red>The shorter diagonal (BD) is approximately 333.453 mm long</font>


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Part (b)


AB = CD = 424
AD = BC = 348


Area = side1*side2*sin(included angle)
Area = AB*BC*sin(angle ABC)
Area = 424*348*sin(130)
<font color=red>Area = 113,031.389671091 square mm approximately</font>


Perimeter = AB+BC+CD+AD
Perimeter = AB+AD+AB+AD
Perimeter = 2*AB + 2*AD
Perimeter = 2*(AB + AD)
Perimeter = 2*(424 + 348)
Perimeter = 2*(772)
<font color=red>Perimeter = 1544 mm</font>
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