Another type of transformation, known as a rotation comes from rotating an image about a fixed point. The fixed point the image is rotated about is known as the center of rotation.

Play with the applet below to explore the rotation transformation. Try changing the shape and size of the original triangle, then use the slider to change the angle of rotation.

Created with Geogebra

The center of rotation does not always have to be a point on the image. Consider the figure below, which shows square $A$A being rotated about the point $O$O.

Square $A$A is rotated $135^\circ$135° clockwise, or $225^\circ$225° counterclockwise, about $O$O resulting in square $B$B.

We can use a protractor to measure the angle of rotation between the original object and the rotated object. We can also use a protractor to measure the correct angle of rotation so we can draw the transformation.

Worked example

Solve: Which is the correct image after triangle $A$A is rotated $90^\circ$90° counterclockwise about the point $O$O?

Think: What point is the image being rotated around and which direction is the image being rotated? We can draw some horizontal and vertical lines to help us visualize the rotation.

Do: First lets draw some horizontal and vertical lines so we can measure the angle of rotation.

Grid split up into four quadrants, each with an angle of $90^\circ$90°.

Since we know that each quadrant has an angle of $90^\circ$90°, all we need to do is rotate the triangle $A$A to the next quadrant in an counterclockwise direction.

Rotating triangle $A$A by $90^\circ$90° counterclockwise around point $O$O leaves us at triangle $D$D, therefore triangle $D$D is the transformed shape.

Reflect: If we were to instead rotate triangle $A$A by $90^\circ$90° clockwise, the correct image would then be triangle $B$B.

Practice questions

question 1

Plot the translation of the point by moving it $11$11 units to the left and $9$9 units down.

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question 2

Which of the following shows the correct plot of the reflection of the triangle across the line $x=-1$x=−1?

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A gray-shaded triangle has its vertices highlighted with blue dots at the coordinates $\left(-5,9\right)$(−5,9), $\left(-2,-1\right)$(−2,−1), and $\left(-8,-5\right)$(−8,−5) on the Cartesian coordinate plane. The axes, labeled "x" and "y," extend from -10 to 10, with major tick marks at intervals of 5 and minor tick marks at intervals of 1. The major tick marks are labeled with numbers to indicate their value on both axes. A vertical line at $x=-1$x=−1 is also plotted, serving as a mirror line for the triangle.

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A gray-shaded triangle has its vertices highlighted with blue dots at the coordinates $\left(-5,-9\right)$(−5,−9), $\left(-2,1\right)$(−2,1), and $\left(-8,5\right)$(−8,5) on the Cartesian coordinate plane. The axes, labeled "x" and "y," extend from -10 to 10, with major tick marks at intervals of 5 and minor tick marks at intervals of 1. The major tick marks are labeled with numbers to indicate their value on both axes. A vertical line at $x=-1$x=−1 is also plotted, serving as a mirror line for the triangle.

A
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A gray-shaded triangle has its vertices highlighted with blue dots at the coordinates $\left(3,9\right)$(3,9), $\left(6,-1\right)$(6,−1), and $\left(0,-5\right)$(0,−5) on the Cartesian coordinate plane. The axes, labeled "x" and "y," extend from -10 to 10, with major tick marks at intervals of 5 and minor tick marks at intervals of 1. The major tick marks are labeled with numbers to indicate their value on both axes. A vertical line at $x=-1$x=−1 is also plotted, serving as a mirror line for the triangle.

B
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A gray-shaded triangle has its vertices highlighted with blue dots at the coordinates $\left(3,-9\right)$(3,−9), $\left(6,1\right)$(6,1), and $\left(0,5\right)$(0,5) on the Cartesian coordinate plane. The axes, labeled "x" and "y," extend from -10 to 10, with major tick marks at intervals of 5 and minor tick marks at intervals of 1. The major tick marks are labeled with numbers to indicate their value on both axes. A vertical line at $x=-1$x=−1 is also plotted, serving as a mirror line for the triangle.

C
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A gray-shaded triangle has its vertices highlighted with blue dots at the coordinates $\left(3,9\right)$(3,9), $\left(0,-1\right)$(0,−1), and $\left(6,-5\right)$(6,−5) on the Cartesian coordinate plane. The axes, labeled "x" and "y," extend from -10 to 10, with major tick marks at intervals of 5 and minor tick marks at intervals of 1. The major tick marks are labeled with numbers to indicate their value on both axes. A vertical line at $x=-1$x=−1 is also plotted, serving as a mirror line for the triangle.

D

question 3

Consider the shape below. What shape is the result of a rotation by $180^\circ$180° clockwise about point $A$A?

A Cartesian coordinate grid with dashed lines forming 12 by 12 squares. A blue irregular hexagon is located on the top left section of the grid. The hexagon is oriented such that two sides are horizontal, and its vertices are aligned with grid intersections. To the right of the hexagon, a green point labeled $A$`A` is positioned near the center of the grid. Point $A$`A` is located at the intersection of the 7th vertical line and the 6th horizontal line when counting from the left and the bottom of the 12 by 12 Cartesian grid. The bottom segment of the hexagon is is located $4$4 units to the left of point $A$`A` and horizontally aligned with it. The closest vertex of the hexagon is approximately $4$4 units to the left of point $A$`A`.

A Cartesian coordinate grid with dashed lines forming 12 by 12 squares. A blue irregular hexagon is located on the top left section of the grid. The blue irregular hexagon is oriented such that two sides are horizontal, and its vertices are aligned with grid intersections. To the right of the blue irregular hexagon, a green point labeled $A$A is positioned near the center of the grid. Point $A$A is located at the intersection of the 7th vertical line and the 6th horizontal line when counting from the left and the bottom of the 12 by 12 Cartesian grid. The bottom segment of the blue hexagon is located $4$4 units to the left of point $A$A and is horizontally aligned with it. The closest vertex of the blue hexagon is approximately $4$4 square units to the left of point $A$A. $A$A purple irregular hexagon is positioned $above$ point $A$A. The blue irregular hexagon and the purple irregular hexagon are similar in size and have the same distance from point $A$A.

A

A Cartesian coordinate grid with dashed lines forming 12 by 12 squares. A blue irregular hexagon is located on the top left section of the grid. The blue irregular hexagon is oriented such that two sides are horizontal, and its vertices are aligned with grid intersections. To the right of the blue irregular hexagon, a green point labeled $A$A is positioned near the center of the grid. Point $A$A is located at the intersection of the 7th vertical line and the 6th horizontal line when counting from the left and the bottom of the 12 by 12 Cartesian grid. The bottom segment of the blue hexagon is located $4$4 units to the left of point $A$A and is horizontally aligned with it. The closest vertex of the blue hexagon is approximately $4$4 square units to the left of point $A$A. A purple irregular hexagon is positioned $below$ point $A$A. The blue irregular hexagon and the purple irregular hexagon are similar in size and have the same distance from point $A$A.

B

A Cartesian coordinate grid with dashed lines forming 12 by 12 squares. A blue irregular hexagon is located on the top left section of the grid. The blue irregular hexagon is oriented such that two sides are horizontal, and its vertices are aligned with grid intersections. To the right of the blue irregular hexagon, a green point labeled $A$A is positioned near the center of the grid. Point $A$A is located at the intersection of the 7th vertical line and the 6th horizontal line when counting from the left and the bottom of the 12 by 12 Cartesian grid. The bottom segment of the blue hexagon is located $4$4 units to the left of point $A$A and is horizontally aligned with it. The closest vertex of the blue hexagon is approximately $4$4 units to the left of point $A$A. A purple irregular hexagon is positioned $to the right of$ point $A$A. The blue irregular hexagon and the purple irregular hexagon are similar in size and have the same distance from point $A$A.

C

A Cartesian coordinate grid with dashed lines forming 12 by 12 squares. A blue irregular hexagon is located on the top left section of the grid. The blue irregular hexagon is oriented such that two sides are horizontal, and its vertices are aligned with grid intersections. To the right of the blue irregular hexagon, a green point labeled $A$A is positioned near the center of the grid. Point $A$A is located at the intersection of the 7th vertical line and the 6th horizontal line when counting from the left and the bottom of the 12 by 12 Cartesian grid. The bottom segment of the blue hexagon is located $4$4 units to the left of point $A$A and is horizontally aligned with it. The closest vertex of the blue hexagon is approximately $4$4 square units to the left of point $A$A. A purple irregular hexagon is positioned $on the upper left of$ point $A$A. The blue irregular hexagon and the purple irregular hexagon are similar in size and have the same distance from point $A$A.

D