Worksheet: Using Determinants to Calculate Areas

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 𝐴𝐡𝐢 with vertices 𝐴(1,4), 𝐡(βˆ’4,5), and 𝐢(βˆ’4,βˆ’5).

Q3:

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

Q4:

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

Q5:

Consider the quadrilateral with vertices 𝐴(1,3),𝐡(4,2),𝐢(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 ||||𝐡𝐢000𝐴𝐢000𝐴𝐡||||=936. If triangle 𝐴𝐡𝐢 is right triangle at 𝐡. It 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 𝐴𝐡𝐢 with vertices 𝐴(1,βˆ’5), 𝐡(βˆ’4,βˆ’2), and 𝐢(5,βˆ’5).

Q15:

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

  • A14
  • B31
  • C312
  • D16
  • E10

Q16:

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

Q17:

Use determinants to calculate the area of the parallelogram with vertices (1,1), (βˆ’4,5), (βˆ’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 (π‘Ž,𝑏),(4+π‘Ž,1+𝑏),(5+π‘Ž,4+𝑏), and (1+π‘Ž,3+𝑏).

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

Q19:

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

Q20:

The unit square is defined as the square with vertices (0,0), (1,0), (1,1), and (0,1). Consider the parallelogram with vertices 𝐴(0,0), 𝐡(4,2), 𝐢(5,5), and 𝐷(1,3).

Explain how the parallelogram can be produced from matrix 𝑀=4123.

  • A𝐷𝐢𝐡𝐴 is the image of the unit square after multiplication by 𝑀.
  • B𝐴𝐢𝐡𝐷is the image of the unit square after multiplication by 𝑀.
  • C𝐴𝐡𝐢𝐷 is the image of the unit square after multiplication by 𝑀.

Write the area of 𝐴𝐡𝐢𝐷 as a determinant and determine its value.

  • Aarea(𝐴𝐡𝐢𝐷)=𝑀=βˆ’10det
  • Barea(𝐴𝐡𝐢𝐷)=𝑀=10det
  • Carea(𝐴𝐡𝐢𝐷)=𝑀=2det
  • Darea(𝐴𝐡𝐢𝐷)=𝑀=14det
  • Earea(𝐴𝐡𝐢𝐷)=𝑀=βˆ’14det

Explain how the parallelogram can be produced from matrix 𝑁=1432.

  • A𝐴𝐢𝐡𝐷 is the image of the unit square after multiplication by 𝑁.
  • B𝐴𝐷𝐢𝐡 is the image of the unit square after multiplication by 𝑁.
  • C𝐷𝐢𝐡𝐴 is the image of the unit square after multiplication by 𝑁.

Write the area of 𝐴𝐡𝐢𝐷 in terms of the matrix 𝑁.

  • Aarea(𝐴𝐡𝐢𝐷)=14|𝑁|det
  • Barea(𝐴𝐡𝐢𝐷)=2|𝑁|det
  • Carea(𝐴𝐡𝐢𝐷)=|𝑁|det
  • Darea(𝐴𝐡𝐢𝐷)=12|𝑁|det
  • Earea(𝐴𝐡𝐢𝐷)=2|𝑁|det

Q21:

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

  • A𝐴(βˆ’3,6), 𝐡(8,βˆ’7), 𝐢(βˆ’3,βˆ’8)
  • B𝐴(βˆ’10,βˆ’6), 𝐡(βˆ’2,1), 𝐢(0,βˆ’9)
  • C𝐴(βˆ’6,4), 𝐡(βˆ’8,4), 𝐢(3,10)
  • D𝐴(βˆ’10,βˆ’4), 𝐡(βˆ’8,βˆ’2), 𝐢(βˆ’5,1)

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