Question

△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?
c) was length preserved during this sequence of transformations? ______ why?
d) would this sequence of transformations be called a rigid transformation? ______ explain.
rectangle abcd plotted at a(4,2), b(8,2), c(8,8) and d(4,8) is to be reflected in the line x = 1 and then dilated by a scale factor of 1/2. (center of dilation is the origin).
a) plot the image of abcd under this sequence of transformations and label it abcd. state coordinates: a = ____ b = __ c = __ d = ____
b) were the angle measures preserved during this sequence of transformations? ______ why?
c) was length preserved during this sequence of transformations? ______ why?
d) would this sequence of transformations be called a rigid transformation? ______ explain.

Explanation:

Step1: Apply translation to $\triangle ABC$

For point $A(4,2)$:
Using the translation rule $(x,y)\to(x - 10,y + 1)$, we have $x=4,y = 2$. Then $x'=4-10=-6$ and $y'=2 + 1=3$. So the translated - point $A_t=(-6,3)$.
For point $B(8,2)$:
$x = 8,y = 2$. Then $x'=8-10=-2$ and $y'=2 + 1=3$. So the translated - point $B_t=(-2,3)$.
For point $C(6,8)$:
$x = 6,y = 8$. Then $x'=6-10=-4$ and $y'=8 + 1=9$. So the translated - point $C_t=(-4,9)$.

Step2: Apply reflection over the $x$ - axis

The rule for reflecting a point $(x,y)$ over the $x$ - axis is $(x,y)\to(x,-y)$.
For $A_t(-6,3)$: After reflection, $A'=(-6,-3)$.
For $B_t(-2,3)$: After reflection, $B'=(-2,-3)$.
For $C_t(-4,9)$: After reflection, $C'=(-4,-9)$.

Step3: Analyze angle - measure preservation

Translation and reflection are rigid transformations. Rigid transformations preserve angle measures. So, the angle measures of $\triangle ABC$ and $\triangle A'B'C'$ are preserved because translations and reflections do not change the shape's orientation in a way that would alter angles.

Step4: Analyze length preservation

Translation and reflection are rigid transformations. Rigid transformations preserve lengths. So, the lengths of the sides of $\triangle ABC$ and $\triangle A'B'C'$ are preserved because these transformations do not stretch or compress the figure.

Step5: Determine if it's a rigid transformation

This sequence of transformations (translation followed by reflection) is a rigid transformation. A rigid transformation is a transformation that preserves distance (lengths of sides) and angle measures. Since both translation and reflection preserve these properties, the combined transformation is rigid.

Answer:

a) $A'=(-6,-3)$, $B'=(-2,-3)$, $C'=(-4,-9)$
b) Yes. Because translation and reflection are rigid - transformations which preserve angle measures.
c) Yes. Because translation and reflection are rigid - transformations which preserve lengths.
d) Yes. Because both translation and reflection preserve lengths and angle measures, which are the properties of rigid transformations.