Lesson Worksheet: Using Determinants to Calculate Areas Mathematics • 10th Grade

In this worksheet, we will practice using determinants to calculate areas of triangles and parallelograms given the coordinates of their vertices.


Use determinants to work out the area of the triangle with vertices (2,2), (4,2), and (0,2) by viewing the triangle as half of a parallelogram.


Find the area of the triangle below using determinants.

  • A18 square units
  • B26 square units
  • C13 square units
  • D36 square units


Use determinants to calculate the area of the parallelogram with vertices (1,1), (4,5), (2,8), and (3,4).


Using determinants, are the points (0,1), 2,12, and (4,0) collinear?

  • ANo
  • BYes


Consider the quadrilateral with vertices 𝐴(1,3),𝐵(4,2),𝐶(4.5,5), and 𝐷(2,6).

By breaking it into two triangles as shown, calculate the area of this quadrilateral using determinants.


Fill in the blank: If the area of a triangle whose vertices are (,0), (6,0), and (0,3) is 9 square units, then =.

  • A6 or 6
  • B0 or 12
  • C0 or 12
  • D12 or 12


Consider a parallelogram in which (1,3), (3,0), and (1,2) are three vertices.

Complete the following: The area of this parallelogram equals square units.


By using determinants, determine which of the following sets of points are collinear.

  • A𝐴(3,6), 𝐵(8,7), 𝐶(3,8)
  • B𝐴(10,6), 𝐵(2,1), 𝐶(0,9)
  • C𝐴(6,4), 𝐵(8,4), 𝐶(3,10)
  • D𝐴(10,4), 𝐵(8,2), 𝐶(5,1)


If the points (𝑥,1), (3,𝑥), and (0,2) are collinear, use determinants to find all the possible values of 𝑥. Round your answers to two decimal places.

  • A1.00 and 3.00
  • B1.08 and 1.08
  • C4.32 and 8.32
  • D3.00 and 3.00
  • E2.16 and 4.16


Given the points 𝐴(1,𝑥), 𝐵(𝑦,1), 𝐶(3,1), and 𝐷(2,5), find the values of 𝑥 and 𝑦 if the area of the triangle 𝐴𝐵𝐶 is 2 square units and the points 𝐵, 𝐶, and 𝐷 are collinear.

  • A𝑥=23 or 1 and 𝑦=83
  • B𝑥=23 or 1 and 𝑦=83
  • C𝑥=17 or 5 and 𝑦=83
  • D𝑥=17 or 5 and 𝑦=83
  • E𝑥=23 and 𝑦=83

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