Question

rigid or not rigid transformations

1. △abc plotted at a(-4,2), b(-7,2) and c(-7,7), is to be translated according to the rule (x,y)→(x + 10,y - 8).

a) plot the image of △abc under this translation and label it △abc. state the new coordinates:
a=
b=
c=
b) was length preserved during this translation?
describe how you made your decision.
c) are translations rigid transformations?
explain.

1. △abc plotted at a(-4,2), b(-7,2) and c(-7,7), is to be reflected over the y - axis.

a) plot the image of △abc under this reflection and label it △abc. state the new coordinates:
a=
b=
c=
b) was length preserved during this reflection?
describe how you made your decision.
c) are reflections rigid transformations?
explain.

1. rectangle abcd plotted at a(3,2), b(6,2), c(6,9) and d(3,9) is to be rotated 90° (center of rotation is the origin).

a) plot the image of abcd under this rotation and label it abcd. state the new coordinates:
a=
b=
c=
d=
b) were the angle measures preserved during this rotation?
describe how you made your decision.

Explanation:

Step1: Find new coordinates for reflection over y - axis

The rule for reflecting a point $(x,y)$ over the $y$-axis is $(x,y)\to(-x,y)$.
For $A(-4,2)$: $A'=(4,2)$
For $B(-7,2)$: $B'=(7,2)$
For $C(-7,7)$: $C'=(7,7)$

Step2: Determine if length is preserved

Lengths are preserved in a reflection because a reflection is a rigid - transformation. It is an isometry, which means that the distance between any two points in the pre - image is the same as the distance between the corresponding two points in the image. We can use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$ to verify. For example, the distance between $A(-4,2)$ and $B(-7,2)$ is $d_{AB}=\sqrt{(-7+4)^2+(2 - 2)^2}=3$. The distance between $A'(4,2)$ and $B'(7,2)$ is $d_{A'B'}=\sqrt{(7 - 4)^2+(2 - 2)^2}=3$.

Step3: Determine if reflection is a rigid transformation

Reflections are rigid transformations. A rigid transformation preserves the size and shape of the figure. In a reflection, the pre - image and the image are congruent. The lengths of the sides and the measures of the angles of the pre - image are equal to the lengths of the corresponding sides and the measures of the corresponding angles of the image.

Step1: Find new coordinates for 90 - degree rotation about the origin

The rule for rotating a point $(x,y)$ 90 degrees counter - clockwise about the origin is $(x,y)\to(-y,x)$.
For $A(3,2)$: $A'=(-2,3)$
For $B(6,2)$: $B'=(-2,6)$
For $C(6,9)$: $C'=(-9,6)$
For $D(3,9)$: $D'=(-9,3)$

Step2: Determine if angle measures are preserved

Angle measures are preserved during a rotation. A rotation is a rigid transformation. Rigid transformations preserve the shape and size of a figure. We know that the angle between two lines in the pre - image is equal to the angle between the corresponding two lines in the image. For example, in a rectangle, all angles are 90 degrees in the pre - image, and after rotation, the angles between the corresponding sides of the rotated rectangle are also 90 degrees.

Answer:

a) $A'=(4,2)$, $B'=(7,2)$, $C'=(7,7)$
b) Yes. Reflections are isometries and preserve distance.
c) Yes. Reflections preserve size and shape (congruence).

For the rotation problem: