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.

Q1:

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.

Q2:

Find the area of the triangle below using determinants.

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

Q3:

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

Q4:

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

  • ANo
  • BYes

Q5:

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.

Q6:

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

Q7:

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.

Q8:

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)

Q9:

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

Q10:

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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