Question

1. △rap plotted at r(1,1), a(2,4) and p(4,2) is to be dilated by a scale factor of 2 (center of dilation is origin).

a) plot the image of △rap under this dilation and label it △rap. state the new coordinates.
r = ______ a = ____ p = ______
b) was length preserved during this translation? ________
if not, how do the lengths of the image compare to the lengths of the pre - image?
c) are dilations rigid transformations? ________
explain.

1. △abc plotted at a(4,2), b(8,2) and c(6,8) is to be translated (x,y)→(x - 10,y + 1) and then reflected over the x - axis.

a) plot the image of △abc under this sequence of transformations and label it △abc. state coordinates.
a = ______ b = ____ c = ______
b) were the angle measures preserved during this sequence of transformations? ________ why?
was length preserved during this sequence of transformations? ________ why?
would this sequence of transformations be called a rigid transformation? explain.

Explanation:

Step1: Find new coordinates for dilation of $\triangle RAP$

For a dilation with a scale - factor $k = 2$ centered at the origin $(0,0)$, the rule is $(x,y)\to(kx,ky)$.
Given $R(1,1)$, then $R'=(2\times1,2\times1)=(2,2)$; given $A(2,4)$, then $A'=(2\times2,2\times4)=(4,8)$; given $P(4,2)$, then $P'=(2\times4,2\times2)=(8,4)$.

Step2: Analyze length preservation in dilation

The length of a line - segment with endpoints $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For a dilation with scale factor $k$, the lengths of the sides of the image are $k$ times the lengths of the sides of the pre - image. Since $k = 2$, length is not preserved. The lengths of the sides of the image are 2 times the lengths of the sides of the pre - image.

Step3: Determine if dilation is a rigid transformation

A rigid transformation preserves distance (length) and angle measures. Since dilation changes the lengths of the sides of a figure, dilations are not rigid transformations.

Step4: Find new coordinates for $\triangle ABC$ transformation

First, apply the translation $(x,y)\to(x - 10,y + 1)$:
For $A(4,2)$: $A_1=(4-10,2 + 1)=(-6,3)$; for $B(8,2)$: $B_1=(8-10,2 + 1)=(-2,3)$; for $C(6,8)$: $C_1=(6-10,8 + 1)=(-4,9)$.
Then, reflect over the $x$ - axis. The rule for reflecting over the $x$ - axis is $(x,y)\to(x,-y)$.
So $A'=(-6,-3)$; $B'=(-2,-3)$; $C'=(-4,-9)$.

Step5: Analyze angle - measure preservation

Translations and reflections are rigid transformations. Rigid transformations preserve angle measures. So, the angle measures were preserved during the sequence of transformations because translations and reflections do not change the shape's angles.

Step6: Analyze length preservation

Translations and reflections are rigid transformations, and rigid transformations preserve length. So, length was preserved during the sequence of transformations because translations and reflections do not change the lengths of the sides of the figure. This sequence of transformations (translation and reflection) is a rigid transformation because both translation and reflection preserve distance and angle measures.

Answer:

a) For $\triangle RAP$: $R'=(2,2)$, $A'=(4,8)$, $P'=(8,4)$; for $\triangle ABC$: $A'=(-6,-3)$, $B'=(-2,-3)$, $C'=(-4,-9)$
b) For $\triangle RAP$: No, lengths of the image are 2 times the lengths of the pre - image; for $\triangle ABC$: Yes, because translations and reflections are rigid transformations.
c) No, because dilations change the lengths of the sides of a figure.
b) (for $\triangle ABC$): Yes, because translations and reflections are rigid transformations which preserve angle measures.
c) (for $\triangle ABC$): Yes, because translations and reflections preserve distance and angle measures.