Vector Algebra Question 172
Question: The position vectors of the vertices of a quadrilateral ABCD are $ a,,b,,c $ and d respectively. Area of the quadrilateral formed by joining the middle points of its sides is
[Roorkee 2000]
Options:
A) $ \frac{1}{4},|a\times b+b\times d+d\times a| $
B) $ \frac{1}{4},| b\times c+c\times d+a\times d+b\times a | $
C) $ \frac{1}{4},| a\times b+b\times c+c\times d+d\times a | $
D) $ \frac{1}{4}|\text{ b }\times\text{ c+c }\times\text{ d+d }\times\text{ b }| $ .
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Answer:
Correct Answer: C
Solution:
- It is given that $ \mathbf{a},\mathbf{b},\mathbf{c} $ and $ \mathbf{d} $ are the position vectors of vertices of a quadrilateral ABCD respectively.
Let E, F, G and H are the middle points of sides AB, BC, CD and DA respectively.
The position vectors of these points will be $ \overrightarrow{OE}=\frac{1}{2}(\mathbf{a}+\mathbf{b}),,\overrightarrow{OF}=\frac{1}{2}(\mathbf{b}+\mathbf{c}) $ ,
$ \overrightarrow{OG}=\frac{1}{2}(\mathbf{c}+\mathbf{d}),,\overrightarrow{OH}=\frac{1}{2}(\mathbf{a}+\mathbf{d}) $
Then $ \overrightarrow{EF}=\overrightarrow{OF}-\overrightarrow{OE}=( \frac{\mathbf{c}-\mathbf{a}}{2} ) $
and $ \overrightarrow{FG}=\frac{1}{2}(\mathbf{d}-\mathbf{b}),\overrightarrow{GH}=\frac{1}{2}(\mathbf{a}-\mathbf{c}),,\overrightarrow{GH}=\frac{1}{2}(\mathbf{b}-\mathbf{d}) $
It is clear that $ \overrightarrow{EF} $ is parallel to $ \overrightarrow{GH} $ and $ \overrightarrow{FG} $ is parallel to $ \overrightarrow{HE} $ . Thus EFGH is a parallelogram.
$ \therefore ,\overrightarrow{EF}\times \overrightarrow{FG}=\frac{1}{4}{(\mathbf{c}-\mathbf{a})\times (\mathbf{d}-\mathbf{b})} $
$ =\frac{1}{4}(\mathbf{c}\times \mathbf{d}-\mathbf{c}\times \mathbf{b}-\mathbf{a}\times \mathbf{d}+\mathbf{a}\times \mathbf{b}) $
$ =\frac{1}{4}(\mathbf{a}\times \mathbf{b}+\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{d}+\mathbf{d}\times \mathbf{a}) $
$ \therefore $
Area of parallelogram EFGH is,
$ A=|\overline{EF}\times \overline{FG}| $ $ =\frac{1}{4}|\mathbf{a}\times \mathbf{b}+\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{d}+\mathbf{d}\times \mathbf{a}| $ .
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