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Question: In this seating arrangement of desks in a classroom of three students Rohini, Sandhya and Bina are s...

In this seating arrangement of desks in a classroom of three students Rohini, Sandhya and Bina are seated at A(3,1),B(6,4) and C(8,6)\text{A}\left( \text{3},\text{1} \right),\text{B}\left( \text{6},\text{4} \right)\text{ and C}\left( \text{8},\text{6} \right) Do you think they are seated in a line?

Explanation

Solution

We will use the concept that if we have three points A, B and C the slope of line AB and BC are equal if the points A, B and C lies in the same line. The slope of a line joining the points (x1,y1) and (x2,y2)\left( {{x}_{1}},{{y}_{1}} \right)\text{ and }\left( {{x}_{2}},{{y}_{2}} \right) is given by,
Slope(m)=y2−y1x2−x1Slope\left( m \right)=\dfrac{y{}_{2}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}

Complete step-by-step solution:
We have been given that there students Rohini, Sandhya and Bina are seated at A(3,1),B(6,4) and C(8,6)\text{A}\left( \text{3},\text{1} \right),\text{B}\left( \text{6},\text{4} \right)\text{ and C}\left( \text{8},\text{6} \right) and asked to find whether they are seated in a line.
We know that if we have three points lying in the same line then the slope of the line taking two points at a time is equal.
Also, the slope of a line joining the points (x1,y1) and (x2,y2)\left( {{x}_{1}},{{y}_{1}} \right)\text{ and }\left( {{x}_{2}},{{y}_{2}} \right) is given by
Slope(m)=y2−y1x2−x1Slope\left( m \right)=\dfrac{y{}_{2}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}
We have A(3,1),B (6,4) and C(8,6)\text{A}\left( \text{3},\text{1} \right),\text{B }\left( \text{6},\text{4} \right)\text{ and C}\left( \text{8},\text{6} \right)
Slope of AB ⇒4−16−3=33=1\Rightarrow \dfrac{4-1}{6-3}=\dfrac{3}{3}=1
Slope of BC ⇒6−48−6=22=1\Rightarrow \dfrac{6-4}{8-6}=\dfrac{2}{2}=1
Slope of AC ⇒6−18−3=55=1\Rightarrow \dfrac{6-1}{8-3}=\dfrac{5}{5}=1
Since the slope of line AB, BC and AC are the same so the points A, B, and C must lie on the same line.
We can also plot the situation graphically as shown below,

Therefore, Rohini, Sandhya, and Bina are seated in a line.

Note: We can also solve the given question by using the concept of distance between the points. If A, B and C are three points lying on the same line then sum of distance AB and BC is equal to AC. We will use distance formula to find distance between the two points. It is given by root d=(x2−x1)2+(y2−y1)2d=\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}} where (x1,y1) and (x2,y2)\left( {{x}_{1}},{{y}_{1}} \right)\text{ and }\left( {{x}_{2}},{{y}_{2}} \right) are coordinates of points.
Also, be careful while using the formula of the slope as there is a chance of a significant mistake.