Look at the drawings below and explain them.

Here are the explanations for the problems shown in the image.

1. Rotations

• a) $\frac{1}{4}$ turn = $90^\circ$: This shows a shape rotated by $90^\circ$ clockwise around its center. A quarter turn is equivalent to a $90^\circ$ rotation.
• b) $\frac{1}{2}$ turn = $180^\circ$: This shows the same shape rotated by $180^\circ$ clockwise around its center. A half turn is equivalent to a $180^\circ$ rotation.

2. Look at the drawings below and explain them.

• Top left: The square is moved horizontally to the right without changing its orientation. This transformation is called a translation.
• Top right: The rectangle is turned around a fixed point (the small circle with an arrow). It has been rotated $90^\circ$ clockwise. This transformation is called a rotation.
• Bottom left: The square is turned around a fixed point. It has been rotated $90^\circ$ counter-clockwise. This transformation is also a rotation.
• Bottom right: The rectangle is flipped over a horizontal line (the dashed line). This transformation is called a reflection.

3. Complete the table below by rotating each shape and draw the rotated shape.

Since I cannot draw, I will describe the orientation of the rotated shapes. Assume rotations are clockwise unless specified.

• Row 1 (Bowtie/Hourglass shape):

  • Original: Two triangles joined at a point, one pointing up, one pointing down.
  • $90^\circ$ rotation: The shape will appear rotated $90^\circ$. The top triangle will now point to the right, and the bottom triangle will point to the left.
  • $180^\circ$ rotation: The shape will look identical to the original, as it has $180^\circ$ rotational symmetry.
  • $270^\circ$ rotation: The shape will appear rotated $270^\circ$ clockwise (or $90^\circ$ counter-clockwise). The top triangle will now point to the left, and the bottom triangle will point to the right.

• Row 2 (Horizontal rectangle with a dot on the top-left corner):

  • Original: A horizontal rectangle with a dot at its top-left vertex.
  • $90^\circ$ rotation: The rectangle will become vertical. The dot will be at the top-right vertex of the new vertical rectangle.
  • $180^\circ$ rotation: The rectangle will be horizontal again. The dot will be at the bottom-right vertex.
  • $270^\circ$ rotation: The rectangle will be vertical again. The dot will be at the bottom-left vertex of the new vertical rectangle.

• Row 3 (Upright triangle with a dot on the bottom-left vertex):

  • Original: An equilateral or isosceles triangle pointing upwards, with a dot at its bottom-left vertex.
  • $90^\circ$ rotation: The triangle will point to the right. The dot will be at the top vertex of the triangle.
  • $180^\circ$ rotation: The triangle will point downwards (upside-down). The dot will be at the top-right vertex.
  • $270^\circ$ rotation: The triangle will point to the left. The dot will be at the bottom vertex of the triangle.