Determinant And Area

Question

The area of the triangle with vertices at the points (1, 2), (4, 6) and (7, 2) in the coordinate plane, computed using the determinant formula, equals which value?

RankGuru · Accepted Answer

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.

Determinant And Area

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