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.

  • A26 square units
  • B13 square units
  • C36 square units
  • D18 square units

Q2:

Find the area of the triangle 𝐴 𝐵 𝐶 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 | | | | 𝐵 𝐶 0 0 0 𝐴 𝐶 0 0 0 𝐴 𝐵 | | | | = 9 3 6 . If triangle 𝐴 𝐵 𝐶 is right triangle at 𝐵 . It has an area of 39, what is the radius of its circumcircle?

  • A12
  • B6
  • C 1 6
  • D 1 2 4
  • E24

Q7:

Find the area of the triangle below using determinants.

  • A17 square units
  • B34 square units
  • C38 square units
  • D19 square units

Q8:

Find the area of the triangle below using determinants.

  • A22 square units
  • B32 square units
  • C16 square units
  • D44 square units

Q9:

Find the area of the triangle below using determinants.

  • A30 square units
  • B13 square units
  • C26 square units
  • D15 square units

Q10:

Find the area of the triangle below using determinants.

  • A20 square units
  • B60 square units
  • C40 square units
  • D30 square units

Q11:

Find the area of the triangle below using determinants.

  • A3 square units
  • B6 square units
  • C36 square units
  • D18 square units

Q12:

Find the area of the triangle below using determinants.

  • A11 square units
  • B8 square units
  • C22 square units
  • D16 square units

Q13:

Find the area of the triangle below using determinants.

  • A20 square units
  • B10 square units
  • C17 square units
  • D34 square units

Q14:

Find the area of the triangle 𝐴 𝐵 𝐶 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 ) .

  • A 3 1 2
  • B14
  • C31
  • 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:

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 ) , 𝐵 ( 4 , 2 ) , 𝐶 ( 5 , 5 ) , and 𝐷 ( 1 , 3 ) .

Explain how the parallelogram can be produced from matrix 𝑀 = 4 1 2 3 .

  • A 𝐷 𝐶 𝐵 𝐴 is the image of the unit square after multiplication by 𝑀 .
  • B 𝐴 𝐶 𝐵 𝐷 is the image of the unit square after multiplication by 𝑀 .
  • C 𝐴 𝐵 𝐶 𝐷 is the image of the unit square after multiplication by 𝑀 .

Write the area of 𝐴 𝐵 𝐶 𝐷 as a determinant and determine its value.

  • Aarea ( 𝐴 𝐵 𝐶 𝐷 ) = 𝑀 = 1 0 d e t
  • Barea ( 𝐴 𝐵 𝐶 𝐷 ) = 𝑀 = 1 4 d e t
  • Carea ( 𝐴 𝐵 𝐶 𝐷 ) = 𝑀 = 1 0 d e t
  • Darea ( 𝐴 𝐵 𝐶 𝐷 ) = 𝑀 = 1 4 d e t
  • Earea ( 𝐴 𝐵 𝐶 𝐷 ) = 𝑀 = 2 d e t

Explain how the parallelogram can be produced from matrix 𝑁 = 1 4 3 2 .

  • A 𝐴 𝐷 𝐶 𝐵 is the image of the unit square after multiplication by 𝑁 .
  • B 𝐷 𝐶 𝐵 𝐴 is the image of the unit square after multiplication by 𝑁 .
  • C 𝐴 𝐶 𝐵 𝐷 is the image of the unit square after multiplication by 𝑁 .

Write the area of 𝐴 𝐵 𝐶 𝐷 in terms of the matrix 𝑁 .

  • Aarea ( 𝐴 𝐵 𝐶 𝐷 ) = 1 4 | 𝑁 | d e t
  • Barea ( 𝐴 𝐵 𝐶 𝐷 ) = 1 2 | 𝑁 | d e t
  • Carea ( 𝐴 𝐵 𝐶 𝐷 ) = 2 | 𝑁 | d e t
  • Darea ( 𝐴 𝐵 𝐶 𝐷 ) = 2 | 𝑁 | d e t
  • Earea ( 𝐴 𝐵 𝐶 𝐷 ) = | 𝑁 | d e t

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