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
To determine the set of transformations performed on triangle XYZ to form triangle X''Y''Z'', we need to compare the coordinates of the corresponding vertices.
Triangle XYZ:
X: (-5, -1)
Y: (-6, -2)
Z: (-3, -2)
Triangle X''Y''Z'':
X'': (10, -2)
Y'': (12, -4)
Z'': (6, -4)
From the given coordinates, we can observe that the x-coordinates have been doubled and the y-coordinates have been multiplied by -2.
Therefore, the set of transformations 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.
Hence, the correct answer is:
Dilation by a scale factor of 2 followed by reflection about the x-axis.
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.
To determine the set of transformations that has been performed on triangle XYZ to form triangle X′Y′Z′, we need to compare the corresponding sides and angles of the two triangles.
First, let's analyze the corresponding sides:
- The length of XY is √((-6) - (-5))² + ((-2) - (-1))² = √2.
- The length of X'Y' is √(12 - 10)² + ((-4) - (-2))² = √8.
- The length of XZ is √((-3) - (-5))² + ((-2) - (-1))² = √5.
- The length of X'Z' is √(6 - 10)² + ((-4) - (-2))² = √20.
- The length of YZ is √((-3) - (-6))² + ((-2) - (-2))² = 3.
- The length of Y'Z' is √(6 - 12)² + ((-4) - (-4))² = 6.
From these calculations, we can see that the lengths of the corresponding sides of the two triangles are not proportional. Therefore, a dilation by a scale factor is not the only transformation performed. We can eliminate options involving just dilation.
Now, let's analyze the corresponding angles:
- ∠XYZ is a right angle.
- ∠X'Y'Z' is a right angle.
From this, we can observe that the corresponding angles of the two triangles are congruent, which indicates that no rotation has occurred.
Considering the information gathered, we can conclude that the transformation performed is a reflection about either the x-axis or the y-axis.
To differentiate between the two options, we can look at the y-coordinates of the vertices:
- The y-coordinate of X is -1.
- The y-coordinate of X'' is -2.
Since the y-coordinates of the vertices are not the same, the transformation is not a reflection about the x-axis. Therefore, the set of transformations that has been performed on triangle XYZ to form triangle X′′Y′′Z′′ is a dilation by a scale factor of 2 followed by reflection about the y-axis.
Therefore, the correct option is: Dilation by a scale factor of 2 followed by reflection about the y-axis.
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
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.