Line segment CD is shown on a coordinate grid:
A coordinate plane is shown. Line segment CD has endpoints 1 comma 2 and 1 comma negative 1.
The line segment is reflected about the y-axis to form C′D′. Which statement describes C′D′? (4 points)
C′D′ is perpendicular to CD.
C′D′ is half the length of CD.
C′D′ is greater than twice the length of CD.
C′D′ and CD are equal in length.
C′D′ is equal in length to CD.
Line segment EF is shown on a coordinate grid:
A coordinate plane is shown. Line segment EF has endpoints negative 3 comma 2 and negative 1 comma 2.
The line segment is rotated 180 degrees counterclockwise about the origin to form E′F′. Which statement describes E′F′? (4 points)
E′F′ is perpendicular to EF.
E′F′ and EF are equal in length.
E′F′ is half the length of EF.
E′F′ is greater than twice the length of EF.
E′F′ and EF are equal in length.
A shape is shown on the graph:
A coordinate plane is shown. Triangle ABC has vertices 3 comma 0, 3 comma negative 3, and 1 comma negative 3.
Which of the following is a reflection of the shape? (4 points)
A coordinate plane is shown. A triangle has vertices 3 comma 0, 1 comma 0, and 3 comma 3.
A coordinate plane is shown. A triangle has vertices 2 comma 4, 2 comma 1 and 4 comma 4.
A coordinate plane is shown. A triangle has vertices negative 1 comma 0, negative 1 comma negative 3, and negative 3 comma negative 3.
A coordinate plane is shown. A triangle has vertices at negative 3 comma 0, negative 3 comma negative 3, and negative 1 comma negative 3.
The reflection of Triangle ABC would have vertices at 3, 0, 3, 3, and 1, 3.
Therefore, the option "A coordinate plane is shown. A triangle has vertices 3, 0, 1, 0, and 3, 3" is the reflection of the shape.
A rotation is shown in the drawing:
A green dart-shaped figure is shown, labeled Figure X. A curved dotted line shows the rotation around to the left to create a blue dart-shaped figure labeled Figure X prime. The new image is directly to the right of the original.
Which statement best describes the rotation? (4 points)
Clockwise 90-degree rotation
Clockwise 270-degree rotation
Counterclockwise 90-degree rotation
Counterclockwise 270-degree rotation
Based on the given information, the rotation described is a counterclockwise 90-degree rotation.
Angle X measures 110 degrees. If angle X is rotated 60 degrees, what is the measure of angle X′? (4 points)
0 degree
60 degrees
110 degrees
170 degrees
If angle X is rotated 60 degrees, the measure of angle X′ will be 110 degrees + 60 degrees = 170 degrees.
Therefore, the correct answer is 170 degrees.
To determine the characteristics of the reflected line segment C′D′, we need to understand the process of reflecting a point or a figure across the y-axis.
When a point is reflected across the y-axis, its x-coordinate is negated while the y-coordinate remains the same. In other words, if a point is given as (x, y), its reflection across the y-axis would be (-x, y).
In the case of line segment CD, the endpoints are (1, 2) and (1, -1). To reflect these points across the y-axis, we negate their x-coordinates. Therefore, the reflected endpoints C′D′ would be (-1, 2) and (-1, -1).
Now, let's analyze the given statements:
1. C′D′ is perpendicular to CD:
This statement cannot be determined just by reflecting the line segment across the y-axis. Reflecting a line segment does not affect its perpendicularity properties.
2. C′D′ is half the length of CD:
By comparing the x-coordinates of the endpoints, we can see that both line segments have the same length. Therefore, this statement is not true.
3. C′D′ is greater than twice the length of CD:
By comparing the x-coordinates of the endpoints, we can see that both line segments have the same length. Therefore, this statement is not true.
4. C′D′ and CD are equal in length:
By comparing the x-coordinates of the endpoints, we can see that both line segments have the same length. Therefore, this statement is true.
Hence, the correct statement is: C′D′ and CD are equal in length.