Determinant And Area

The area of a triangle with given vertices is half the absolute value of a determinant formed from the coordinates, linking matrices and coordinate geometry in JEE Advanced. The area equals (1/2)|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|. Substituting the vertices (1,2),(4,6),(7,2): area = (1/2)|1(6 - 2) + 4(2 - 2) + 7(2 - 6)| = (1/2)|1·4 + 4·0 + 7·(-4)| = (1/2)|4 + 0 - 28| = (1/2)|-24| = 12. Option 6 forgets the absolute value scaling or halves twice. Option 24 omits the factor of one half. Option 10 miscomputes a coordinate difference. Hence the area is 12. Plausibility check: the base from (1,2) to (7,2) has length 6 along the line y = 2, and the apex (4,6) is at height 4 above it, giving area (1/2)·6·4 = 12, matching the determinant computation exactly. The determinant formula for triangle area is the two-dimensional shadow of the general principle that determinants measure signed volume spanned by vectors, with the sign encoding orientation. The same construction detects collinearity, since three collinear points yield a zero determinant and therefore zero area, making this single formula serve double duty across coordinate-geometry questions in the syllabus.