Question

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? yes 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.
c) are rotations rigid transformations? explain.

Explanation:

Step1: Apply reflection rule

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

Step2: Analyze length preservation

Lengths are preserved in a reflection because reflection is a rigid - transformation. The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. The pre - image and image points have the same perpendicular distance from the line of reflection, so lengths are the same.

Step3: Define rigid transformation

A rigid transformation preserves distance and angle measures. Reflections preserve distance (lengths of sides of a figure) and angle measures, so reflections are rigid transformations.

Step4: Apply rotation rule

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

Step5: Analyze angle preservation in rotation

Rotations are rigid transformations. In a rotation, the shape and size of the figure remain the same. The relative orientation of the sides changes, but the angle measures between the sides remain the same because the transformation is isometric (preserves distances and angles).

Step6: Confirm rotation as rigid transformation

Rotations preserve distance (lengths of sides) and angle measures. The transformation is based on rotating the figure around a fixed point (the origin in this case) without stretching or compressing it, so rotations are rigid transformations.

Answer:

1. a) $A'=(4,2)$, $B'=(7,2)$, $C'=(7,7)$

b) Yes. Reflections are rigid transformations that preserve distance, and the perpendicular distance from the line of reflection for pre - image and image points is the same.
c) Yes. Reflections preserve distance and angle measures, which are the characteristics of rigid transformations.

1. a) $A'=(-2,3)$, $B'=(-2,6)$, $C'=(-9,6)$, $D'=(-9,3)$

b) Yes. Rotations are rigid transformations that preserve angle measures as the shape and size of the figure remain the same.
c) Yes. Rotations preserve distance and angle measures, which are the defining features of rigid transformations.