# 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 with vertices , , and .

Q3:

Use determinants to find the area of the triangle with vertices , , and .

Q4:

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

Q5:

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

Q6:

Consider the equation If triangle is right triangle at . 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 with vertices , , and .

Q15:

Use determinants to calculate the area of the quadrilateral with vertices , , , and .

• A14
• B31
• C
• D16
• E10

Q16:

Use determinants to calculate the area of the polygon with vertices , , , , and .

Q17:

Use determinants to calculate the area of the parallelogram with vertices , , , and .

Q18:

Use determinants to calculate the area of the parallelogram with vertices , and .

Use determinants to calculate the area of the parallelogram with vertices , and .

Use determinants to calculate the area of the parallelogram with vertices , and .

Q19:

Use determinants to calculate the area of the polygon with vertices , , , , and .

Q20:

The unit square is defined as the square with vertices , , , and . Consider the parallelogram with vertices , , , and .

Explain how the parallelogram can be produced from matrix .

• A is the image of the unit square after multiplication by .
• Bis the image of the unit square after multiplication by .
• C is the image of the unit square after multiplication by .

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

• Aarea
• Barea
• Carea
• Darea
• Earea

Explain how the parallelogram can be produced from matrix .

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

Write the area of in terms of the matrix .

• Aarea
• Barea
• Carea
• Darea
• Earea

Q21:

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

• A, ,
• B, ,
• C, ,
• D, ,