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Question
The area of the quadrilateral ABCD, where A(0,4,1), B(2, 3, –1), C(4, 5, 0) and D(2, 6, 2), is equal to ______.
Options
9 sq.units
18 sq.units
27 sq.units
81 sq.units
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Solution
The area of the quadrilateral ABCD, where A(0,4,1), B(2, 3, –1), C(4, 5, 0) and D(2, 6, 2), is equal to 9 sq.units.
Explanation:
Given points are A(0, 4, 1), B(2,3,– 1), C(4, 5, 0) and D(2,6,2)
D’ratios of AB = 2,–1 –2
And d’ratios of DC = 2,–1,–2
∴ AB||DC
Similarly, d’ratios of AD = 2, 2, 1 and d’ratios of BC = 2, 2, 1
∴ AD || BC
So ABCD is a parallelogram
`vec"AB" = 2hat"i" - hat"j" - 2hat"k"`
`vec"AD" = 2hat"i" + 2hat"j" + hat"k"`
∴ Area of parallelogram ABCD = `|vec"AB" xx vec"AD"|`
= `|(hat"i", hat"j", hat"k"),(2, -1, -2),(2, 2, 1)|`
= `hat"i"(-1 + 4) - hat"j"(2 + 4) + hat"k"(4 + 2)`
= `3hat"i" - 6hat"j" + 6hat"k"`
= `sqrt((3)^2 + (-6)^2 + (6)^2)`
= `sqrt(9 + 36 + 36)`
= `sqrt(81)`
= 9 sq.units
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