Angle of Rotation

Chapter 49

Angle of Rotation Concept Explanation:

The Angle of Rotation refers to the measure of the degree through which a point, line, or shape is rotated about a fixed point, called the center of rotation. The rotation can be clockwise or counterclockwise. When rotating figures on the coordinate plane, the center of rotation is typically the origin $(0, 0)$ .

An angle of rotation is usually measured in degrees, where a complete circle represents 360°, a half-turn is 180°, and a quarter-turn is 90°. Rotations follow specific rules depending on the angle:

90° Counterclockwise Rotation:

$(x, y) \rightarrow (-y, x)$

180° Rotation (either clockwise or counterclockwise):

$(x, y) \rightarrow (-x, -y)$

270° Counterclockwise Rotation (which is the same as 90° clockwise):

$(x, y) \rightarrow (y, -x)$

Example:

Consider rotating the point $A(2, 3)$ about the origin:

90° Counterclockwise Rotation:

$A(2, 3) \rightarrow (-3, 2)$

180° Rotation:

$A(2, 3) \rightarrow (-2, -3)$

270° Counterclockwise Rotation:

$A(2, 3) \rightarrow (3, -2)$

Common Mistakes:

Wrong Direction of Rotation: Students often confuse clockwise and counterclockwise rotations, which can lead to incorrect transformations.
Not Applying the Rotation Rule Correctly: Sometimes the wrong formula is used for rotation, especially in 90° and 270° rotations.
Misidentifying the Center of Rotation: Students may mistakenly assume the origin is always the center of rotation. This can lead to incorrect results when the center of rotation is elsewhere.

Helpful Tips:

Visualize the Rotation: Sketch the point and its image to ensure you’re rotating in the correct direction.
Memorize the Rotation Rules: Understanding the specific transformation rules for different angles will help avoid errors.
Double-Check the Angle: Confirm whether the angle is clockwise or counterclockwise before applying the rule.

Diagrams:

Here’s an example of rotating a point on the coordinate plane.

*** QuickLaTeX cannot compile formula:

\begin{tikzpicture}
\draw<a href="-3,-3">help lines</a> grid (3,3);
\draw<a href="-3.5,0">-></a> -- (3.5,0) node[right] {x};
\draw<a href="0,-3.5">-></a> -- (0,3.5) node[above] {y};
\draw<a href="2,3">fill=black</a> circle (2pt) node[above right] {A(2,3)};
\draw<a href="-3,2">fill=black</a> circle (2pt) node[above left] {A'(-3,2)};
\draw<a href="2,3">->, thick</a> -- (-3,2) node[midway, above] {90° CCW};
\end{tikzpicture}


*** Error message:
Error: Cannot create svg file

The diagram shows the point $A(2,3)$ being rotated 90° counterclockwise to become $A'(-3,2)$ .

Hard Questions:

Q1: Rotate the point $P(4, 1)$ about the origin by 90° counterclockwise. What are the coordinates of the new point?

Step-by-Step Solution:

Apply the 90° counterclockwise rotation rule:

$(x, y) \rightarrow (-y, x)$

Substitute the coordinates of $P(4, 1)$ :

$(4, 1) \rightarrow (-1, 4)$

Answer:

The new coordinates of $P$ are $(-1, 4)$ .

Q2: Rotate the point $Q(-5, 2)$ about the origin by 180°. What are the new coordinates of $Q$ ?

Step-by-Step Solution:

Apply the 180° rotation rule:

$(x, y) \rightarrow (-x, -y)$

Substitute the coordinates of $Q(-5, 2)$ :

$(-5, 2) \rightarrow (5, -2)$

Answer:

The new coordinates of $Q$ are $(5, -2)$ .

Q3: Rotate the point $R(3, -4)$ about the origin by 270° counterclockwise. What are the new coordinates of $R$ ?

Step-by-Step Solution:

Apply the 270° counterclockwise (or 90° clockwise) rotation rule:

$(x, y) \rightarrow (y, -x)$

Substitute the coordinates of $R(3, -4)$ :

$(3, -4) \rightarrow (-4, -3)$

Answer:

The new coordinates of $R$ are $(-4, -3)$ .

Q4: A point $T(-3, -7)$ is rotated 90° clockwise. Find the coordinates of the new point.

Step-by-Step Solution:

A 90° clockwise rotation is the same as a 270° counterclockwise rotation, so apply the rule:

$(x, y) \rightarrow (y, -x)$

Substitute the coordinates of $T(-3, -7)$ :

$(-3, -7) \rightarrow (-7, 3)$

Answer:

The new coordinates of $T$ are $(-7, 3)$ .

Q5: If a triangle with vertices $A(1, 2)$ , $B(3, 4)$ , and $C(5, 2)$ is rotated 180° about the origin, what are the coordinates of the new vertices?

Step-by-Step Solution:

Apply the 180° rotation rule to each vertex:

$(x, y) \rightarrow (-x, -y)$

For $A(1, 2)$ :

$(1, 2) \rightarrow (-1, -2)$

For $B(3, 4)$ :

$(3, 4) \rightarrow (-3, -4)$

For $C(5, 2)$ :

$(5, 2) \rightarrow (-5, -2)$

Answer:

The new vertices of the triangle are $A'(-1, -2)$ , $B'(-3, -4)$ , and $C'(-5, -2)$ .

Avoiding Common Mistakes:

Rotation Direction: Always confirm whether you are rotating clockwise or counterclockwise. A 90° clockwise rotation is the same as a 270° counterclockwise rotation.
Center of Rotation: Unless explicitly stated, the center of rotation is assumed to be the origin. If the center is different, use the rules of coordinate transformation relative to that point.
Multiple Rotations: When performing multiple rotations, apply each rotation step-by-step to avoid errors.

Helpful Links for Further Study:

Learn about Rotations and Transformations: A comprehensive tutorial on geometric rotations.
Interactive Rotations on Desmos: Use Desmos to practice rotating points and shapes on the coordinate plane.

In summary, the Angle of Rotation is a foundational concept in geometry that allows us to move points and figures around a fixed point by a specified angle. Understanding the rotation rules and avoiding common mistakes will help you master this concept easily.

Toll Free: 1 (888) 356-0607