G7b - Rotation - BossMaths.com - Free Printable

Let's solve this step-by-step.

We are given a shape (a triangle) on a coordinate grid and asked to rotate it about the origin $(0, 0)$ in three different ways:

- (a) $90^\circ$ clockwise
- (b) $90^\circ$ anticlockwise
- (c) $180^\circ$

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Step 1: Identify the coordinates of the vertices

From the graph, the blue triangle has three vertices. Let's find their coordinates:

- The bottom-left vertex is at $(2, 1)$
- The top-left vertex is at $(2, 5)$
- The right tip (pointy end) is at $(4, 3)$

So the three points are:
- $A = (2, 1)$
- $B = (2, 5)$
- $C = (4, 3)$

We will apply rotation rules to each point.

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Rotation Rules About the Origin

For a point $(x, y)$:

- 90° clockwise: $(x, y) \rightarrow (y, -x)$
- 90° anticlockwise: $(x, y) \rightarrow (-y, x)$
- 180°: $(x, y) \rightarrow (-x, -y)$

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## (a) Rotate 90° clockwise about (0, 0)

Apply rule: $(x, y) \rightarrow (y, -x)$

- $A = (2, 1) \rightarrow (1, -2)$
- $B = (2, 5) \rightarrow (5, -2)$
- $C = (4, 3) \rightarrow (3, -4)$

So the new vertices are:
- $(1, -2)$
- $(5, -2)$
- $(3, -4)$

This triangle will be in the fourth quadrant, pointing downward to the right.

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## (b) Rotate 90° anticlockwise about (0, 0)

Apply rule: $(x, y) \rightarrow (-y, x)$

- $A = (2, 1) \rightarrow (-1, 2)$
- $B = (2, 5) \rightarrow (-5, 2)$
- $C = (4, 3) \rightarrow (-3, 4)$

New vertices:
- $(-1, 2)$
- $(-5, 2)$
- $(-3, 4)$

This triangle will be in the second quadrant, pointing upward to the left.

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## (c) Rotate 180° about (0, 0)

Apply rule: $(x, y) \rightarrow (-x, -y)$

- $A = (2, 1) \rightarrow (-2, -1)$
- $B = (2, 5) \rightarrow (-2, -5)$
- $C = (4, 3) \rightarrow (-4, -3)$

New vertices:
- $(-2, -1)$
- $(-2, -5)$
- $(-4, -3)$

This triangle will be in the third quadrant, upside down and mirrored across the origin.

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✔ Summary of Rotations:

| Original Point | 90° Clockwise | 90° Anticlockwise | 180° |
|----------------|---------------|--------------------|--------|
| (2, 1) | (1, -2) | (-1, 2) | (-2, -1) |
| (2, 5) | (5, -2) | (-5, 2) | (-2, -5) |
| (4, 3) | (3, -4) | (-3, 4) | (-4, -3) |

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🔍 How to Draw Them?

You can now plot these transformed points on the same grid:

- For (a): Plot (1, -2), (5, -2), (3, -4)
- For (b): Plot (-1, 2), (-5, 2), (-3, 4)
- For (c): Plot (-2, -1), (-2, -5), (-4, -3)

Connect the points to form triangles.

> 🦆 Note: The duck family at the top is just decorative — no math there!

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Final Answer:

(a) After $90^\circ$ clockwise:
Vertices: $(1, -2),\ (5, -2),\ (3, -4)$

(b) After $90^\circ$ anticlockwise:
Vertices: $(-1, 2),\ (-5, 2),\ (-3, 4)$

(c) After $180^\circ$:
Vertices: $(-2, -1),\ (-2, -5),\ (-4, -3)$

These are the rotated images of the original triangle.