SOLUTION: In parallelogram ABCD, AB = 13, AD = 14, and the length of diagonal AC is 15. What is the area of the parallelogram in square units?

Algebra ->  Parallelograms -> SOLUTION: In parallelogram ABCD, AB = 13, AD = 14, and the length of diagonal AC is 15. What is the area of the parallelogram in square units?      Log On


   

Question 1190321: In parallelogram ABCD, AB = 13, AD = 14, and the length of diagonal AC is 15. What is the area of the parallelogram in square units?
Answer by ikleyn(52855) About Me  (Show Source):
You can put this solution on YOUR website!
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In parallelogram ABCD, AB = 13, AD = 14, and the length of diagonal AC is 15.
What is the area of the parallelogram in square units?
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Apply the cosine law and find cosine of the angle B between the sides AB and BC of the parallelogram

(the diagonal AC is the opposite side of the triangle ABC to angle B)


    AC^2 = AB^2 + BC^2 - 2*AB*BC*cos(B)

    15^2 = 13^2 + 14^2 - 2*13*14*cos(B)

    cos(B) = %2813%5E2+%2B+14%5E2+-+15%5E2%29%2F%282%2A13%2A14%29 = 140%2F%282%2A13%2A14%29 = 10%2F26.


Next, find sin(B) = sqrt%281-cos%5E2%28B%29%29= sqrt%281+-+%2810%2F26%29%5E2%29 = 24%2F26 = 12%2F13.


Finally, find the area of the parallelogram ABCD


    area%5BABCD%5D = AB*BC*sin(B) = 13%2A14%2A%2812%2F13%29 = 14*12 = 168 square units.    ANSWER

Solved.     //     An amazing phenomenon is that this answer is a precise integer number  ( ! )

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Another way to solve the problem is to find the area of the triangle  ABC  using
the  Heron's formula  (the lengths of the sides of the triangle are given  ( ! ) ).

After finding the area of the triangle  ABC  simply double it,
and you will get the area of the parallelogram  ABCD.