Question

Which describes how triangle FGH could be transformed to triangle F prime G prime H prime in two steps?

On a coordinate plane, triangle F G H has points (negative 2, 1), (negative 3, 3), (0, 1). Triangle F prime G prime H prime has points (negative 8, negative 4), (negative 12, negative 12), (0, negative 4).

Answer 1

Translation: Shift the entire triangle 6 units to the left and 5 units down.

Dilation: Enlarge the triangle by a factor of 4.

Explanation:

To describe how triangle FGH can be transformed to triangle F'G'H' in two steps, we need to identify the specific transformations applied.

Translation:

The first step involves a translation, where the entire triangle is shifted by a certain amount horizontally and vertically. To determine the translation vector, we subtract the coordinates of corresponding vertices from F'G'H' from those of FGH.

Translation vector = (x-coordinate difference, y-coordinate difference) = ((-8) - (-2), (-4) - 1) = (-6, -5)

So, the translation vector is (-6, -5), indicating that the triangle is shifted 6 units to the left and 5 units down.

Dilation:

The second step involves a dilation, which changes the size of the triangle. To determine the dilation factor, we can compare the lengths of corresponding sides.

The length of side FG is given by the distance formula:

FG = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((-3 - (-2))^2 + (3 - 1)^2) = sqrt(1^2 + 2^2) = sqrt(5)

The length of side F'G' is given by the distance formula as well:

F'G' = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((-12 - (-8))^2 + (-12 - (-4))^2) = sqrt(4^2 + 8^2) = sqrt(80) = 4sqrt(5)

The dilation factor is the ratio of the corresponding side lengths:

Dilation factor = F'G' / FG = (4sqrt(5)) / sqrt(5) = 4

The dilation factor is 4, indicating that the triangle is enlarged by a factor of 4.