Question

Find the area of the triangle formed by the lines  \(y -x = 0, x + y = 0\) and \(x - k = 0.\)

Answer 1

The equations of the given lines are 
\(y - x = 0 … (1) \)
\(x + y = 0 … (2) \)
\(x - k = 0 … (3) \)
The point of intersection of lines (1) and (2) is given by x = 0 and y = 0 
The point of intersection of lines (2) and (3) is given by x = k and y = -k 
The point of intersection of lines (3) and (1) is given by x = k and y = k 
Thus, the vertices of the triangle formed by the three given lines are (0, 0), (k, -k), and (k, k).

We know that the area of a triangle whose vertices are  \( (x_1, y_1), (x_2, y_2),\) and \((x_3, y_3)\) is
\(\frac{1}{2} |x_1 (y_2 – y_3) + x_2 (y_3 – y_1) + x_3 (y_1 – y_2)|\)

Therefore, area of the triangle formed by the three given lines 
\(= \frac{1}{2} |0 (-k – k) + k (k – 0) + k (0 + k)|\) square units

\(= \frac{1}{2}|k^2 + k^2| \)square units

\(= \frac{1}{2}|2k^2|\) square units

\(=k^2\)square units