Finding the Area of a Parallelogram from Coordinates

We are given the coordinates of three consecutive vertices of a parallelogram ABCD as A(1, 3), B(-1, 2), and C(3, 5). We need to find the area of this parallelogram.

A common method to find the area of a parallelogram when given the coordinates of its vertices is to use vector properties. If we consider two adjacent sides of the parallelogram originating from the same vertex, say vectors \(\vec{u}\) and \(\vec{v}\), the area of the parallelogram is the magnitude of the determinant of the matrix formed by their components in 2D.

Since A, B, and C are consecutive vertices, we can consider the vectors representing two sides originating from vertex B: \(\vec{BA}\) and \(\vec{BC}\).

• The coordinates of A are (1, 3) and B are (-1, 2). The vector \(\vec{BA}\) is calculated as \(A - B\). \[\vec{BA} = (1 - (-1), 3 - 2) = (1 + 1, 1) = (2, 1)\] So, the components of \(\vec{BA}\) are \(u_x = 2\) and \(u_y = 1\).
• The coordinates of B are (-1, 2) and C are (3, 5). The vector \(\vec{BC}\) is calculated as \(C - B\). \[\vec{BC} = (3 - (-1), 5 - 2) = (3 + 1, 3) = (4, 3)\] So, the components of \(\vec{BC}\) are \(v_x = 4\) and \(v_y = 3\).

The area of the parallelogram formed by vectors \(\vec{u} = (u_x, u_y)\) and \(\vec{v} = (v_x, v_y)\) is given by the absolute value of the expression \(u_x v_y - u_y v_x\). This is equivalent to the magnitude of the 2D cross product (if viewed in 3D). Area = \(|u_x v_y - u_y v_x|\)

Using the components of \(\vec{BA} = (2, 1)\) and \(\vec{BC} = (4, 3)\):

The area of the parallelogram ABCD is 2 square units.

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