Question

Two similar triangles are shown on the coordinate grid:

A coordinate plane is shown. Triangle XYZ has vertices X at negative 5 comma negative 1, Y at negative 6 comma negative 2, and Z at negative 3 comma negative 2. Triangle X double-prime Y double-prime Z double-prime has vertices X double-prime at 10 comma negative 2, Y double-prime at 12 comma negative 4, and Z double-prime at 6 comma negative 4.
Which set of transformations has been performed on triangle XYZ to form triangle X′′Y′′Z′′? (4 points)


Dilation by a scale factor of 4 followed by reflection about the x-axis

Dilation by a scale factor of 2 followed by reflection about the y-axis

Dilation by a scale factor of 4 followed by reflection about the y-axis

Dilation by a scale factor of 2 followed by reflection about the x-axis

Answer 1

To determine which set of transformations has been performed on triangle XYZ to form triangle X''Y''Z'', we can compare the corresponding coordinates of the vertices.

First, let's compare the x-coordinates:
X = -5, X'' = 10
Y = -6, Y'' = 12
Z = -3, Z'' = 6

We can see that the x-coordinates have been multiplied by a scale factor of 2 (X'' = -5 * 2, Y'' = -6 * 2, Z'' = -3 * 2). Therefore, there is a dilation by a scale factor of 2.

Next, let's compare the y-coordinates:
X = -1, X'' = -2
Y = -2, Y'' = -4
Z = -2, Z'' = -4

We can see that the y-coordinates have been multiplied by a scale factor of 2 (X'' = -1 * 2, Y'' = -2 * 2, Z'' = -2 * 2). Therefore, there is a dilation by a scale factor of 2.

Finally, let's consider the reflections. Since the signs of the y-coordinates have changed (Y = -2, Y'' = -4), it indicates a reflection about the x-axis. Additionally, since the signs of the x-coordinates have remained the same (X = -5, X'' = -2), there is no reflection about the y-axis.

Based on these comparisons, the set of transformations that has been performed on triangle XYZ to form triangle X''Y''Z'' is: Dilation by a scale factor of 2 followed by reflection about the x-axis. Therefore, the correct answer is:

Dilation by a scale factor of 2 followed by reflection about the x-axis.

Answer 2

Triangle PQR is transformed to similar triangle P′Q′R′:

A coordinate plane is shown. Triangle PQR has vertices P at negative 6 comma 6, Q at negative 6 comma 3, and R at negative 3 comma 3. Triangle P prime Q prime R prime has vertices at P prime negative 2 comma 2, Q prime at negative 2 comma 1, and R prime at negative 1 comma 1.
What is the scale factor of dilation? (4 points)


1 over 2

1 over 3

1 over 4

1 over 5

Answer 3

To find the scale factor of dilation, we can compare the corresponding sides of the two similar triangles.

Triangle PQR:
Length of PQ: sqrt((3-6)^2 + (3-6)^2) = sqrt(18) = 3sqrt(2)
Length of QR: sqrt((3-(-3))^2 + (3-3)^2) = sqrt(36) = 6
Length of RP: sqrt((3-(-6))^2 + (3-6)^2) = sqrt(90) = 3sqrt(10)

Triangle P'Q'R':
Length of P'Q': sqrt((1-2)^2 + (1-2)^2) = sqrt(2)
Length of Q'R': sqrt((1-1)^2 + (1-2)^2) = sqrt(1) = 1
Length of R'P': sqrt((1-(-2))^2 + (1-2)^2) = sqrt(18) = 3sqrt(2)

To find the scale factor of dilation, we can divide the lengths of corresponding sides of the two triangles:

PQR/P'Q'R'
PQ/P'Q' = (3sqrt(2))/(sqrt(2)) = 3
QR/Q'R' = 6/1 = 6
RP/R'P' = (3sqrt(10))/(3sqrt(2)) = sqrt(5)/sqrt(2) = sqrt(5/2)

The scale factor of dilation is equal to the scale factor of any corresponding pair of sides. In this case, the scale factor is 3.

Hence, the correct answer is 1 over 3.