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Find the area of a parallelogram A B C D...

Find the area of a parallelogram `A B C D` if three of its vertices are `A(2,\ 4),\ B(2+sqrt(3),\ 5)` and `C(2,\ 6)` .

A

`(2sqrt(3))`

B

`(3sqrt(4))`

C

`(4sqrt(5))`

D

`(6sqrt(5))`

Text Solution

Verified by Experts

The correct Answer is:
A
Let ABCD be the given parallelogram, there of whose vertices are A(2, 4), B(2+`sqrt(3),` 5) and C(2, 6).

In `Delta` ABC, we have
`(x_(1) =2, y_(1) = 4), (x_(2) = 2+sqrt(3), y_(2) = 5), (x_(3) = 2, y_(3) = 6).`
`therefore ar(Delta ABC) = (1)/(2)|(x_(1)(y_(2)-y_(3)) +x_(2)(y_(3)-y_(1)) + x_(3)(y_(1) - y_(2))|`
`=(1)/(2)|2(5-6) +2(2+sqrt(3)(6-4) +2 (4-5)|`
`= (1)/(2)|2 xx (-1) + (4+2sqrt(3))-2|`
`=(1)/(2)(2sqrt(3)) = sqrt(3)` sq unit
`rArr ar("llgm"ABCD) = 2 xx ar(Delta ABC) = (2 xx sqrt(3))` sq units
`=(2sqrt(3))` sq units.
Hence, the area of the given parallelogram is `(2sqrt(3))` sq units.
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