**Graphing rotations :**

To rotate a figure in the coordinate plane, rotate each of its vertices. Then connect the vertices to form the image.

**Example 1 :**

The triangle XYZ has the following vertices X(0, 0), Y(2, 0) and Z(2, 4). Rotate the triangle XYZ 90° counterclockwise about the origin.

**Solution : **

**Step 1 :**

Trace triangle XYZ and the x- and y-axes onto a piece of paper.

**Step 2 :**

**Let X', Y' and Z' be the vertices of the rotated figure.**

Since the triangle is rotated 90° counterclockwise about the origin, the rule is

(x, y) ------> (-y, x)

**Step 3 :**

**X(0, 0) **------> X'(0, 0)

**Y(2, 0) **------> Y'(0, 2)

**Z(2, 4) **------> Z'(-4, 2)

**Step 4 : **

Sketch the image X'Y'Z' using the points X'(0, 0), Y'(0, 2) and Z'(-4, 2).

**Example 2 :**

The triangle PQR has the following vertices P(0, 0), Q(-2, 3) and R(2,3). Rotate the triangle PQR 90° clockwise about the origin.

**Solution : **

**Step 1 :**

Trace triangle PQR and the x- and y-axes onto a piece of paper.

**Step 2 :**

**Let P', Q' and R' be the vertices of the rotated figure.**

Since the triangle is rotated 90° clockwise about the origin, the rule is

(x, y) ------> (y, -x)

**Step 3 :**

**P(0, 0) **------> P'(0, 0)

**Q(-2, 3) **------> Q'(3, 2)

**R(2, 3) **------> R'(3, -2)

**Step 4 : **

Sketch the image P'Q'R' using the points P'(0, 0), Q'(3, 2) and Z'(3, -2).

**Example 3 :**

A quadrilateral has the following vertices A(0, 0), B(1, 2), C(4, 2) and D(3, 0). Rotate the quadrilateral 180° clockwise about the origin.

**Solution : **

**Step 1 :**

Trace the quadrilateral ABCD and the x- and y-axes onto a piece of paper.

**Step 2 :**

**Let A', B', C' and D' be the vertices of the rotated figure.**

Since the quadrilateral is rotated 180° clockwise about the origin, the rule is

(x, y) ------> (-x, -y)

**Step 3 :**

**A(0, 0) **------> A'(0, 0)

**B(1, 2) **------> B'(-1, -2)

**C(4, 2) **------> C'(-4, -2)

**D(3, 0) **------> D'(-3, 0)

**Step 4 : **

Sketch the image A'B'C'D' using the points A'(0, 0), B'(-1, -2), C(-4, -2) and D'(-3, 0).

**Example 4 :**

The triangle XYZ has the following vertices X(0, 0), Y(2, 0) and Z(2, 4). Rotate the triangle XYZ 270° counterclockwise about the origin.

**Solution : **

**Step 1 :**

Trace triangle XYZ and the x- and y-axes onto a piece of paper.

**Step 2 :**

**Let X", Y" and Z" be the vertices of the rotated figure.**

Since the triangle is rotated 270° counterclockwise about the origin, the rule is

(x, y) ------> (y, -x)

**Step 3 :**

**X(0, 0) **------> X"(0, 0)

**Y(2, 0) **------> Y"(0, -2)

**Z(2, 4) **------> Z"(4, -2)

**Step 4 : **

Sketch the image X"Y"Z" using the points X"(0, 0), Y"(0, -2) and Z"(4, -2).

After having gone through the stuff given above, we hope that the students would have understood "Graphing rotations".

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