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In the seating arrangement of desks in a classroom three students Rohini, Sandhya and Bina are seated at $A (3, 1), B (6, 4)$ and $C (8, 6)$. Do you think they are seated in a line?
Given:
Given points are $A (3, 1), B (6, 4)$ and $C (8, 6)$.
To do:
We have to find if Rohini, Sandhya and Bina are seated in a line.
Solution:
If Rohini, Sandhya and Bina are seated in a line then, the points $A, B, C$ should be collinear.
We know that,
The distance between two points \( \mathrm{A}\left(x_{1}, y_{1}\right) \) and \( \mathrm{B}\left(x_{2}, y_{2}\right) \) is \( \sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}} \).
Therefore,
\( \mathrm{AB}=\sqrt{(6-3)^{2}+(4-1)^{2}} \)
\( =\sqrt{(6-3)^{2}+(4-1)^{2}} \)
\( =\sqrt{(3)^{2}+(3)^{2}} \)
\( =\sqrt{9+9} \)
\( =\sqrt{18} \)
\( =3\sqrt{2} \)
\( \mathrm{BC}=\sqrt{(8-6)^{2}+(6-4)^{2}} \)
\( =\sqrt{(2)^{2}+(2)^{2}} \)
\( =\sqrt{4+4} \)
\( =\sqrt{8} \)
\( =2\sqrt{2} \)
\( \mathrm{CA}=\sqrt{(3-8)^{2}+(1-6)^{2}} \)
\( =\sqrt{(-5)^{2}+(-5)^{2}} \)
\( =\sqrt{25+25} \)
\( =\sqrt{50} \)
\( =5\sqrt{2} \)
Here,
\( \mathrm{AB}+\mathrm{BC}=3 \sqrt{2}+2 \sqrt{2}=5 \sqrt{2} \)
\( \mathrm{CA}=5\sqrt{2} \)
\( \mathrm{AB}+\mathrm{BC}=\mathrm{CA} \)
This implies,
\( \mathrm{A}, \mathrm{B} \) and \( \mathrm{C} \) are collinear points.
Hence, they are seated in a line.
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