Q.If the points $(x, -2),\ (5, 2),\ (8, 8)$ are collinear, then $x$ is equal to

For a triangle with vertices $(x_1,y_1),(x_2,y_2),(x_3,y_3)$, the familiar shoelace-type area formula

can be written compactly as a determinant:

The absolute value is taken because the raw determinant can come out negative depending on the (clockwise/anticlockwise) order in which the vertices are listed, whereas area is always non-negative.

Collinearity test. Three points are collinear (lie on one straight line) precisely when the triangle they would form has zero area, i.e. …