Transformations and similarity unit test
Unit 3 lesson 11
1. The transformation T: (x, y) --> (x + 3, y - 2) is a translation. Is the translation a reflection? Explain your answer.
No, the translation is not a reflection. A reflection is a transformation that maintains the same shape and size of an object, but flips it over a line. A translation, on the other hand, moves an object without changing its shape or orientation. In this case, the translation T moves every point (x, y) on the coordinate plane three units to the right and two units down.
2. Given a triangle with vertices A(2, 3), B(3, 4), and C(4, 2), determine the coordinates of the triangle after a reflection over the x-axis.
To reflect a point over the x-axis, we keep the x-coordinate the same, but change the sign of the y-coordinate. So, the coordinates of the reflected triangle would be:
A'(2, -3)
B'(3, -4)
C'(4, -2)
3. Determine the scale factor and coordinates of the figure resulting from a dilation with a scale factor of 1/2 centered at the origin for the figure with vertices P(4, 6), Q(7, 9), and R(10, 6).
To dilate a point with a scale factor of 1/2 centered at the origin, we multiply both the x and y coordinates of the point by 1/2. So, the coordinates of the dilated figure would be:
P'(2, 3)
Q'(3.5, 4.5)
R'(5, 3)
The scale factor can be determined by comparing the distances between the corresponding points. In this case, the scale factor is 1/2, meaning that the dilated figure is half the size of the original figure.
4. Is a dilation a type of transformation that preserves length and angle measures? Explain your answer.
No, a dilation is a type of transformation that changes both the size and shape of an object. It preserves the shape of an object but changes its size by a constant factor. While a dilation can enlarge or shrink an object, it does not preserve length or angle measures. The lengths of the sides of a dilated figure are proportional to the corresponding sides of the original figure, but the actual length values are different. Similarly, the angles of a dilated figure are proportional to the corresponding angles of the original figure, but the actual angle measures are different.
5. Given triangle ABC with vertices A(1, 2), B(2, 4), and C(3, 0), determine the coordinates of the image after a rotation of 90 degrees counterclockwise about the origin.
To rotate a point counterclockwise by 90 degrees about the origin, we switch the x and y coordinates and then negate the new x coordinate. So, the coordinates of the rotated triangle would be:
A'(-2, 1)
B'(-4, 2)
C'(0, 3)
Alternatively, we can use the rotation matrix for a counterclockwise rotation of 90 degrees about the origin:
[x'] [0 -1] [x]
[y'] = [1 0] * [y]
Plugging in the coordinates of each point, we get:
A'(-2, 1)
B'(-4, 2)
C'(0, 3)
6. Which transformation preserves both length and angle measures?
A rigid transformation preserves both length and angle measures. Rigid transformations include translations, rotations, and reflections. These transformations do not change the size or shape of an object, but only move or mirror it in the plane. Lengths of sides and angle measures remain the same after a rigid transformation.