Rotation 90 Degrees Around The Origin

🔑 Key Concepts

• A \(90^\circ\) rotation turns a figure one-quarter turn around the origin.
• For a \(90^\circ\) clockwise rotation, use the rule \((x,y)\rightarrow(y,-x)\).
• For a \(90^\circ\) counterclockwise rotation, use the rule \((x,y)\rightarrow(-y,x)\).
• The \(x\)- and \(y\)-values switch places during a \(90^\circ\) rotation.
• The signs of the new coordinates depend on the quadrant where the image lands.
• A rotation is a rigid transformation because the image keeps the same size and shape.

✏️ Worked Examples

Visual Reference

Apply the \(90^\circ\) Clockwise Rule

Rotate triangle \(ABC\) \(90^\circ\) clockwise around the origin.

Original coordinates:

\(A(4,3)\), \(B(4,5)\), \(C(10,5)\)

Use the clockwise rotation rule:

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

The \(x\)- and \(y\)-values switch places. Then the new \(y\)-value becomes the opposite of the original \(x\)-value.

\(A(4,3)\rightarrow A'(3,-4)\)

\(B(4,5)\rightarrow B'(5,-4)\)

\(C(10,5)\rightarrow C'(5,-10)\)

---

New coordinates:
\(A'=(3,-4)\), \(B'=(5,-4)\), \(C'=(5,-10)\)

What to Notice

The original triangle starts in Quadrant I, where coordinates are positive-positive:

\((+,+)\)

After a \(90^\circ\) clockwise rotation around the origin, the image lands in Quadrant IV, where coordinates are positive-negative:

\((+,-)\)

That is why the image points have positive \(x\)-values and negative \(y\)-values.

A good mental check is to picture the positive \(x\)-axis rotating clockwise. It lands on the negative \(y\)-axis, which helps explain why the signs change.

Right Angle Check

A \(90^\circ\) rotation means the turn from the preimage point to the image point forms a right angle around the origin.

For example, connect:

\(A(4,3)\rightarrow(0,0)\rightarrow A'(3,-4)\)

The two segments meet at the origin and form a \(90^\circ\) angle.

This is a useful way to check whether the figure was actually rotated \(90^\circ\), rather than reflected or translated.

The image keeps the same size and shape, but it turns around the origin.

🧠 Math Vocabulary

• Rotation: a transformation that turns a figure around a fixed point.
• Center of Rotation: the fixed point that a figure rotates around.
• Origin: the point \((0,0)\), where the \(x\)-axis and \(y\)-axis intersect.
• \(90^\circ\) Rotation: a quarter-turn around the center of rotation.
• Clockwise: turning in the same direction as the hands of a clock.
• Counterclockwise: turning in the opposite direction of the hands of a clock.
• Quadrant: one of the four regions of the coordinate plane.
• Rigid Transformation: a transformation that preserves size and shape.
• Image: the new figure after a transformation.
• Preimage: the original figure before a transformation.
• Coordinate Rule: an algebraic rule that shows how ordered pairs change during a transformation.

💡 Main Idea

A \(90^\circ\) rotation around the origin turns a figure one-quarter turn. The coordinates switch places, and the signs change based on whether the rotation is clockwise or counterclockwise.

📚 What You Should Already Know

Students should know how to plot ordered pairs, identify the origin, name the four quadrants, and recognize the sign patterns in each quadrant.

🚀 What Comes Next

After practicing \(90^\circ\) clockwise rotations, students can compare clockwise and counterclockwise rotations, \(180^\circ\) rotations, and other rigid transformations such as reflections and translations.

🧩 Embed This Video in Your LMS

Teachers can embed this YouTube video directly into Canvas, Google Classroom, Schoology, or another learning platform. Click below to show the embed code.