Question

Circle 1 was dilated with the origin as the center of dilation to create Circle 2. Which rule best represents the dilation applied to circle 1 to create circle 2?

a) (x, y) -> (2x, 2y)
b) (x, y) -> (x/2, y/2)
c) (x, y) -> (x + 2, y + 2)
d) (x, y) -> (x - 2, y - 2)

Answer 1

The dilation rule that doubles the size of Circle 1 to create Circle 2 with the origin as the center is (x, y) -> (2x, 2y).

Step-by-step explanation:

The student is asking which rule represents the dilation applied to Circle 1 to create Circle 2 where the origin is the center of dilation. The correct answer is option (a), which is the rule (x, y) -> (2x, 2y). This represents a dilation that doubles the coordinates of any point from Circle 1, resulting in a Circle 2 that is twice as large in radius since the center of dilation is the origin of the coordinate system.

The chosen rule (x, y) -> (2x, 2y) accurately reflects a dilation with a scale factor of 2 applied to Circle 1 to create Circle 2, considering the origin as the center of dilation. This transformation effectively doubles the coordinates of each point from Circle 1, resulting in an enlarged Circle 2 with twice the radius. This understanding aligns with the geometric concept of dilation.

Answer 2

Circle II is larger and appears to be exactly twice the size of Circle I, the correct rule that represents the dilation is:

a) (x, y) -> (2x, 2y).

To determine which rule best represents the dilation applied to Circle 1 to create Circle 2, we need to compare the radii of the two circles since the dilation will affect the size of the circle while keeping its shape.

From the graph, we can visually inspect the coordinates that lie on each circle to determine their radii. By identifying points that lie on the circles and are on the axes, we can easily read off the radius of each circle.

For Circle I (the smaller circle), the radius appears to be half the length of the radius of Circle II (the larger circle), since the graph is grid-based and each square represents one unit.

The rules provided are as follows:

a) (x, y) -> (2x, 2y): This would double the size of the object, which corresponds to what appears to have happened going from Circle I to Circle II.

b) (x, y) -> (x/2, y/2): This would halve the size of the object, which is not what we observe.

c) (x, y) -> (x + 2, y + 2): This would translate the object, not dilate it.

d) (x, y) -> (x - 2, y - 2): This would also translate the object, not dilate it.

Since Circle II is larger and appears to be exactly twice the size of Circle I, the correct rule that represents the dilation is:

a) (x, y) -> (2x, 2y).

This means each coordinate point (x, y) of Circle I is multiplied by 2 to get the corresponding point on Circle II, effectively doubling the radius and creating Circle II.