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.
