Question

What is the rule for the following transformation? 100 points (grade 8, geometry)

Answer 1

Translation: 3 units right and 7 units down.

Explanation:

The mapping rule for a rotation of 90° counter-clockwise about the origin is:

• (x, y) → (-y, x)

The mapping rule for a dilation of 0.25 about the origin is:

• (x, y) → (0.25x, 0.25y)

The mapping rule for a translation of 3 units right and 7 units down is:

• (x, y) → (x+3, y-7)

The mapping rule for a reflection across the y-axis is:

• (x, y) → (-x, y)

To determine the rule that transforms KLMN to K'L'M'N', take one of the vertices from the pre-image and compare to its corresponding vertex in the image.

K = (-3, 4)

K' = (0, -3)

As the numerical values of the x and y coordinates have not be swapped or made negative, the transformation cannot be a rotation of 90 degrees about the origin, or a reflection in the y-axis.

As the x and y coordinates of K' are not 0.25 times the x and y coordinates of K, then the transformation cannot be a dilation of 0.25 about the origin.

Therefore, the transformation that transforms KLMN to K'L'M'N' must be:

• translation of 3 units right and 7 units down.

To check, apply the mapping rule (x, y) → (x+3, y-7) to the vertices of KLMN:

• K = (-3, 4) → K' = (-3+3, 4-7) = (0, -3)
• L = (-3, 5) → L' = (-3+3, 5-7) = (0, -2)
• M = (1, 5) → M' = (1+3, 5-7) = (4, -2)
• N = (1, 4) → N' = (1+3, 4-7) = (4, -3)

Therefore, this confirms that the transformation is a translation of 3 units right and 7 units down.