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ABCD is a parallelogram with side AB=12 cm. Its diagonal AC and BD are of lengths 20 cm and 16 cm respectively. Find the area of parallelogram ABCD.

I tried,

Area of parallelogram=1/2*product of diagonals =1/2*20*16

But, this is not matching the answer at the back of the book.

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  • $\begingroup$ This: Area of parallelogram=1/2*product of diagonals is wrong. A rectangle is a special case of a parallelogram, so the equation should apply to it, too, right? But if you take a rectangle 1 by 100, its diagonals are about 100, so the expression above gives area about 5000, while it actually is 100... $\endgroup$ – CiaPan Jul 10 '14 at 12:19
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For a parallelogram of diagonals $p$ and $q$ the area is actually given by $$A=\frac{pq}{2}\sin(\theta)$$ given that $\theta$ is the angle between the diagonals. The formula you used is for a rhombus which is a special case of the parallelogram wherein all the four sides are equal.

Hint. To solve this problem you can see here that a parallelogram can be 'formed' into a rectangle with the same area. From this we can gather the side length $BC$ and use the formula for a rectangle, namely $A=b\times h.$

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  • $\begingroup$ I am not quite familiar with logarithms. Is there any other method to solve the problem. $\endgroup$ – Abhishekstudent Jul 10 '14 at 12:27
  • $\begingroup$ There were no references to logarithms in my answer. $\endgroup$ – user138999 Jul 10 '14 at 12:28
  • $\begingroup$ What does that sine-theta means? $\endgroup$ – Abhishekstudent Jul 10 '14 at 12:31
  • $\begingroup$ That the the trigonometric sine function. Are you not familiar with trigonometry? $\endgroup$ – user138999 Jul 10 '14 at 12:34
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    $\begingroup$ Ok! Thanks! Once I complete trigonometry chapter, I will attempt this question. $\endgroup$ – Abhishekstudent Jul 10 '14 at 12:39
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I suggest to use the J.Finnegan answer, but add a Law of cosines to it, and when you have a cosine of the angle between diagonals use Pythagorean trigonometric identity to obtain its sine.

EDIT
You can also use a Heron's formula to calculate an area of a triangle with sides 12, 10 and 8 cm (hope ypu know which triangle it is), then find its height. The height doubled becomes a height of the whole parallelogram, so you can easily make a final step.

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