Fig. 7.6 shows the arrangement of desks in a classroom. Ashima, Bharti and Camella are seated at A(3, 1), B(6, 4) and C(8, 6) respectively. Do you think they are seated in a line? Give reasons for your answer. Fig. 7.6
Step-by-Step Solution
Key Concept: Three points are collinear if the slope of the line joining any two pairs of points is the same (or equivalently, the area of the triangle formed by them is zero).
1. Write down the coordinates of the three points: \(A(3,1),\; B(6,4),\; C(8,6)\).
2. Compute the slope of \(AB\):
$$m_{AB}=\frac{4-1}{6-3}=\frac{3}{3}=1$$
3. Compute the slope of \(BC\):
$$m_{BC}=\frac{6-4}{8-6}=\frac{2}{2}=1$$
4. Since \(m_{AB}=m_{BC}=1\), the two slopes are equal; therefore the three points lie on the same straight line (they are collinear).
*Alternative method*: Find the area of triangle \(ABC\) using the determinant formula:
$$\Delta = \frac{1}{2}\begin{vmatrix}3&1&1\\6&4&1\\8&6&1\end{vmatrix}=0$$
The area being zero also confirms collinearity.
5. Hence, Ashima, Bharti and Camella are seated in a straight line.
Correct Answer: Yes, the three points are collinear because the slopes \(AB\) and \(BC\) are equal (both equal to 1), or equivalently, the area of triangle \(ABC\) is zero.