SOLUTION: The sides of a parallelogram ABCD are AB= 424mm and AD= 348mm If angle ABC is 130° a. Find the lengths of long and short diagonals. b. Find the area of and perimeter of a p

Question 1196783: The sides of a parallelogram ABCD are AB= 424mm and AD= 348mm
If angle ABC is 130°

a. Find the lengths of long and short diagonals.
b. Find the area of and perimeter of a parallelogram
Found 2 solutions by math_tutor2020, Theo:
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

Part (a)

Diagram:

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
The longer diagonal (AC) is approximately 700.406 mm long

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
The shorter diagonal (BD) is approximately 333.453 mm long

=============================================================================================

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)
Area = 113,031.389671091 square mm approximately

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)
Perimeter = 1544 mm

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
i was able to solve this using the law of cosines.
that law says:
c^2 = a^2 + b^2 - 2*a*b*cos(C)
what i got was:
long diagonal = 700.4064496911
short diagonal = 333.4528530858
i verified using an online parallelogram calculator found at https://www.calculatorsoup.com/calculators/geometry-plane/parallelogram.php
it did not solve it directly, so i used one of my answer for the long diagonal to have it solve for the other properties of the parallelogram and they checked out.
here's what the calculator showed me after i gave it the two side length that were given as part of the problem and the long diagonal that i calculated using the law of cosines.

i'll be available to answer any questions you have about this.

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