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If diagonals of a parallelogram are straight P with rightwards arrow on top space and space straight Q with rightwards arrow on top comma then find the area of the parallelogram.


Let the sides of the parallelogram be straight A with rightwards arrow on top space and space straight B with rightwards arrow on top, whose diagonals are straight P with rightwards arrow on top space and space stack straight Q. with rightwards arrow on top 

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equals space open vertical bar fraction numerator straight P with rightwards harpoon with barb upwards on top plus straight Q with rightwards harpoon with barb upwards on top over denominator 2 end fraction cross times fraction numerator straight P with rightwards harpoon with barb upwards on top minus straight Q with rightwards harpoon with barb upwards on top over denominator 2 end fraction close vertical bar space

equals space 1 fourth open vertical bar open parentheses straight P with rightwards harpoon with barb upwards on top plus straight Q with rightwards harpoon with barb upwards on top close parentheses cross times open parentheses straight P with rightwards harpoon with barb upwards on top space minus space straight Q with rightwards harpoon with barb upwards on top close parentheses close vertical bar

equals space 1 fourth open vertical bar open parentheses straight P with rightwards harpoon with barb upwards on top cross times straight P with rightwards harpoon with barb upwards on top space plus space straight Q with rightwards harpoon with barb upwards on top cross times straight P with rightwards harpoon with barb upwards on top space minus space straight P with rightwards harpoon with barb upwards on top space cross times straight Q with rightwards harpoon with barb upwards on top space minus space straight Q with rightwards harpoon with barb upwards on top space cross times space straight Q with rightwards harpoon with barb upwards on top close parentheses close vertical bar

equals space 1 fourth open vertical bar open parentheses straight Q with rightwards harpoon with barb upwards on top cross times straight P with rightwards harpoon with barb upwards on top space plus straight Q with rightwards harpoon with barb upwards on top cross times straight P with rightwards harpoon with barb upwards on top close parentheses space space close vertical bar

equals space 1 half space open vertical bar straight Q with rightwards harpoon with barb upwards on top cross times straight P with rightwards harpoon with barb upwards on top close vertical bar space

equals space 1 half open vertical bar straight P with rightwards harpoon with barb upwards on top space cross times straight Q with rightwards harpoon with barb upwards on top close vertical bar

 Hence, the area of the parallelogram is found. 

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