Question

lunging lizard: rotation
rotate the lizard 90° about the point (12,7) so it looks like the lizard is diving into the puddle of mud.
strategies:
definition of rotation:

Explanation:

To solve the rotation of the lizard \( 90^\circ \) about the point \( (12,7) \), we follow these steps for a general point \( (x,y) \) on the lizard:

Step 1: Translate the Point

Subtract the coordinates of the center of rotation \( (12,7) \) from the point \( (x,y) \) to move the center to the origin.
Let the original point be \( (x,y) \). The translated point \( (x', y') \) is:
\[
x' = x - 12, \quad y' = y - 7
\]

Step 2: Rotate \( 90^\circ \) Counterclockwise (or Clockwise, Depending on Direction)

For a \( 90^\circ \) counterclockwise rotation about the origin, the transformation is \( (x', y') \to (-y', x') \).
For a \( 90^\circ \) clockwise rotation, the transformation is \( (x', y') \to (y', -x') \).

Since the lizard should “dive into the mud,” we assume a clockwise rotation (so the lizard points downward). Thus:
\[
x'' = y', \quad y'' = -x'
\]

Step 3: Translate Back

Add the coordinates of the center \( (12,7) \) back to the rotated point \( (x'', y'') \) to return to the original coordinate system:
\[
x_{\text{final}} = x'' + 12, \quad y_{\text{final}} = y'' + 7
\]

Step 4: Apply to Each Vertex of the Lizard

Repeat Steps 1–3 for every labeled point (A, B, C, D, E, F, G, H) of the lizard. For example, take point A (approximate coordinates \( (12, 12) \)):

1. Translate: \( x' = 12 - 12 = 0 \), \( y' = 12 - 7 = 5 \)
2. Rotate clockwise: \( x'' = 5 \), \( y'' = -0 = 0 \)
3. Translate back: \( x_{\text{final}} = 5 + 12 = 17 \), \( y_{\text{final}} = 0 + 7 = 7 \)

Step 5: Draw the Rotated Lizard

Plot all rotated vertices and connect them to form the lizard diving into the mud (the lower puddle).

Final Result

The lizard is rotated \( 90^\circ \) clockwise about \( (12,7) \), with each vertex transformed using the above steps. The rotated lizard will appear to point downward toward the mud puddle.

Answer:

To solve the rotation of the lizard \( 90^\circ \) about the point \( (12,7) \), we follow these steps for a general point \( (x,y) \) on the lizard:

Step 1: Translate the Point

Subtract the coordinates of the center of rotation \( (12,7) \) from the point \( (x,y) \) to move the center to the origin.
Let the original point be \( (x,y) \). The translated point \( (x', y') \) is:
\[
x' = x - 12, \quad y' = y - 7
\]

Step 2: Rotate \( 90^\circ \) Counterclockwise (or Clockwise, Depending on Direction)

For a \( 90^\circ \) counterclockwise rotation about the origin, the transformation is \( (x', y') \to (-y', x') \).
For a \( 90^\circ \) clockwise rotation, the transformation is \( (x', y') \to (y', -x') \).

Since the lizard should “dive into the mud,” we assume a clockwise rotation (so the lizard points downward). Thus:
\[
x'' = y', \quad y'' = -x'
\]

Step 3: Translate Back

Add the coordinates of the center \( (12,7) \) back to the rotated point \( (x'', y'') \) to return to the original coordinate system:
\[
x_{\text{final}} = x'' + 12, \quad y_{\text{final}} = y'' + 7
\]

Step 4: Apply to Each Vertex of the Lizard

Repeat Steps 1–3 for every labeled point (A, B, C, D, E, F, G, H) of the lizard. For example, take point A (approximate coordinates \( (12, 12) \)):

1. Translate: \( x' = 12 - 12 = 0 \), \( y' = 12 - 7 = 5 \)
2. Rotate clockwise: \( x'' = 5 \), \( y'' = -0 = 0 \)
3. Translate back: \( x_{\text{final}} = 5 + 12 = 17 \), \( y_{\text{final}} = 0 + 7 = 7 \)

Step 5: Draw the Rotated Lizard

Plot all rotated vertices and connect them to form the lizard diving into the mud (the lower puddle).

Final Result

The lizard is rotated \( 90^\circ \) clockwise about \( (12,7) \), with each vertex transformed using the above steps. The rotated lizard will appear to point downward toward the mud puddle.