# Lesson Worksheet: Linear Transformations in Planes: Reflection Mathematics • 10th Grade

In this worksheet, we will practice finding the matrix of linear transformation of reflection along the x- or y-axis or the line of a given equation and the image of a vector under the reflection.

Q1:

Which of the following are necessary and sufficient conditions on , , , and for the matrix to represent a reflection?

• A, ,
• B, ,
• C, ,
• D, ,
• E, ,

Q2:

A reflection in a line through the origin sends the vector to . Find the matrix representation of this reflection.

• A
• B
• C
• D
• E

Q3:

A reflection in a line through the origin sends the vector to . Find the matrix representation of this reflection.

• A
• B
• C
• D
• E

Q4:

Consider the reflection in the line .

Find the matrix that represents this transformation.

• A
• B
• C
• D
• E

What is the image of the point under this reflection?

• A
• B
• C
• D
• E

Q5:

Consider the linear transformation which maps a point to its reflection in the -axis.

Find the matrix which represents this transformation.

• A
• B
• C
• D
• E

Where does this transformation map the point ?

• A
• B
• C
• D
• E

Q6:

Consider the given figure. The points , , , and are corners of the unit square. This square is reflected in the line with equation to form the image .

As is the image of in the line through and , . Use this fact and the identity to find the gradient and hence equation of from the gradient of .

• A
• B
• C
• D
• E

Using the fact that is perpendicular to , find the equation of .

• A
• B
• C
• D
• E

Using the fact that , find the coordinates of and .

• A,
• B,
• C,
• D,
• E,

Using the fact that a reflection in a line through the origin is a linear transformation, find the matrix which represents reflection in the line .

• A
• B
• C
• D
• E

Q7:

Consider the matrix where .

Find .

• A
• B
• C
• D
• E

Find .

• A2
• B1
• C
• D0
• E

By drawing the image of the unit square under the transformation, identify the geometrical transformation this matrix corresponds to.

• Aa projection onto the line
• Ba rotation by clockwise about the point
• Ca reflection in the line
• Da rotation of clockwise about the origin
• Ea reflection in the line

Q8:

Consider the given figure. The points , , , and are corners of the unit square. This square is reflected in the line with equation to form the image .

As is the image of in the line through and , . Use this fact and the identity to find the gradient and hence equation of from the gradient of .

• A
• B
• C
• D
• E

Using the fact that is perpendicular to , find the equation of .

• A
• B
• C
• D
• E

Using the fact that , find the coordinates of and .

• A,
• B,
• C,
• D,
• E,

Using the fact that a reflection in a line through the origin is a linear transformation, find the matrix which represents reflection in the line .

• A
• B
• C
• D
• E