Lesson Worksheet: Using Determinants to Calculate Areas Mathematics

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

Q1:

Find the area of the triangle below using determinants.

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

Q2:

Find the area of the triangle 𝐴𝐵𝐶 with vertices 𝐴(1,4), 𝐵(−4,5), and 𝐶(−4,−5).

Q3:

Use determinants to find the area of the triangle with vertices (0,−1), (0,2), and (5,0).

Q4:

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.

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:

Consider the equation ||||𝐵𝐶000𝐴𝐶000𝐴𝐵||||=936. If triangle 𝐴𝐵𝐶 is a right triangle at 𝐵 and has an area of 39, what is the radius of its circumcircle?

Q7:

Find the area of the triangle below using determinants.

  • A19 square units
  • B34 square units
  • C17 square units
  • D38 square units

Q8:

Find the area of the triangle below using determinants.

  • A16 square units
  • B44 square units
  • C22 square units
  • D32 square units

Q9:

Find the area of the triangle below using determinants.

  • A15 square units
  • B26 square units
  • C13 square units
  • D30 square units

Q10:

Find the area of the triangle below using determinants.

  • A20 square units
  • B60 square units
  • C30 square units
  • D40 square units

Q11:

Find the area of the triangle below using determinants.

  • A18 square units
  • B6 square units
  • C3 square units
  • D36 square units

Q12:

Find the area of the triangle below using determinants.

  • A11 square units
  • B16 square units
  • C8 square units
  • D22 square units

Q13:

Find the area of the triangle below using determinants.

  • A17 square units
  • B20 square units
  • C10 square units
  • D34 square units

Q14:

Find the area of the triangle 𝐴𝐵𝐶 with vertices 𝐴(1,−5), 𝐵(−4,−2), and 𝐶(5,−5).

Q15:

Use determinants to calculate the area of the quadrilateral with vertices (−1,−1), (3,−2), (4,1), and (0,3).

  • A14
  • B31
  • C312
  • D16
  • E10

Q16:

Use determinants to calculate the area of the polygon with vertices (−1,−1), (1,−2), (3,1), (0,3), and (−2,1).

Q17:

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

Q18:

Use determinants to calculate the area of the parallelogram with vertices (0,0),(4,1),(5,4), and (1,3).

Use determinants to calculate the area of the parallelogram with vertices (𝑎,𝑏),(4+𝑎,1+𝑏),(5+𝑎,4+𝑏), and (1+𝑎,3+𝑏).

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

Q19:

Use determinants to calculate the area of the polygon with vertices (0,0), (2,−1), (4,2), (1,4), and (−1,2).

Q20:

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)

Q21:

If the area of the triangle with vertices at (𝑥,3), (𝑦,0), and (𝑧,3) is 3, use determinants to find the value of |𝑧−𝑥|.

Q22:

Consider a triangle 𝐴𝐵𝐶 in which 𝑀(6,3), 𝑁(4,2), and 𝐿(5,0) are the midpoints of its sides. Use determinants to find the area of triangle 𝐴𝐵𝐶.

  • A10 square units
  • B7.5 square units
  • C15 square units
  • D5 square units
  • E20 square units

Q23:

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.

Q24:

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

  • ANo
  • BYes

Q25:

If the area of the triangle with vertices (1,ℎ), (0,3), and (4,−5) equals triple the area of the parallelogram in which the points (1,2), (ℎ,0), and (3,4) are vertices, find all possible values of ℎ.

  • A−2
  • B−5 and −0.8
  • C−2 and 2
  • D−2 and −0.5
  • E2

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