Question Video: Calculating the Area of Quadrilateral Using Determinants of Matrices Mathematics

Video Transcript

Consider the quadrilateral with vertices 𝐴 one, three; 𝐵 four, two; 𝐶 4.5, five; and 𝐷 two, six. By breaking it into two triangles as shown, calculate the area of this quadrilateral using determinants.

We want to find the area of this quadrilateral using the areas of the two triangles and determinants. We can start by recalling that the area of a triangle is given by half the absolute value of the determinant of the matrix whose rows are given by each coordinate of a vertex and a final component of one. We can apply this to each triangle separately.

Let’s start with 𝑇 sub one, which is triangle 𝐴𝐶𝐷. We substitute the given coordinates of these points into the formula to obtain the following expression. We can then expand over the first row of the matrix to obtain the following expression, which we can calculate is equal to 4.25 square units.

We need to apply this same process for the other triangle. We can do this by substituting the coordinates of 𝐴, 𝐵, and 𝐶 into the formula in any order to obtain the following expression. We can then expand over the first row and evaluate to find that the area of triangle 𝐴𝐵𝐶 is 4.75 square units. Finally, we can find the area of the quadrilateral by adding the areas of the two triangles. We find that it has an area of nine square units.