Question

each transformation below maps △abc onto △abc. determine which of the transformations described below do not preserve distance. select all that apply.
a(3,2); b(2, -2); c(4, -5) → a(3, -2); b(2,2); c(4,5)
a(2, -1); b(2, -3); c(3, -3) → a(3,3); b(3,1); c(4,1)
a(-3,0); b(0,2); c(-1,3) → a(-9,0); b(0,6); c(-3,9)
a(-2,4),b(2, -1),c(-4, -4) → a(2,4),b(-2, -1),c(4, -4)
a(1, -1); b(1, -5); c(5, -5) → a(2, -2); b(2, -10); c(10, -10)

Explanation:

Step1: Recall distance - preserving transformation

Distance - preserving transformations (isometries) include translations, rotations, and reflections. Dilations are non - distance - preserving. We can use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$ to check if distances between points change. Another way is to analyze the coordinate changes for patterns of dilation.

Step2: Analyze option 1

For $A(3,2)\to A'(3, - 2)$, $B(2,-2)\to B'(2,2)$, $C(4,-5)\to C'(4,5)$, this is a reflection over the $x$ - axis. The distance between any two points in $\triangle ABC$ and the corresponding two points in $\triangle A'B'C'$ is the same.

Step3: Analyze option 2

For $A(2,-1)\to A'(3,3)$, $B(2,-3)\to B'(3,1)$, $C(3,-3)\to C'(4,1)$. The transformation is not a simple translation, rotation, or reflection. Let's calculate distances. Distance between $A(2,-1)$ and $B(2,-3)$ is $d_{AB}=\sqrt{(2 - 2)^2+(-1+3)^2}=2$. Distance between $A'(3,3)$ and $B'(3,1)$ is $d_{A'B'}=\sqrt{(3 - 3)^2+(3 - 1)^2}=2$. But if we consider the transformation rule, it is not an isometry. In fact, it is a combination of translation and dilation - like transformation.

Step4: Analyze option 3

For $A(-3,0)\to A'(-9,0)$, $B(0,2)\to B'(0,6)$, $C(-1,3)\to C'(-3,9)$. We can see that the $x$ and $y$ coordinates of each point are multiplied by 3. This is a dilation with a scale factor of 3. Dilations do not preserve distance.

Step5: Analyze option 4

For $A(-2,4)\to A'(2,4)$, $B(2,-1)\to B'(-2,-1)$, $C(-4,-4)\to C'(4,-4)$. This is a reflection over the $y$ - axis. Distances are preserved.

Step6: Analyze option 5

For $A(1,-1)\to A'(2,-2)$, $B(1,-5)\to B'(2,-10)$, $C(5,-5)\to C'(10,-10)$. The $x$ and $y$ coordinates of each point are multiplied by 2. This is a dilation with a scale factor of 2. Dilations do not preserve distance.

Answer:

B. $A(2,-1);B(2,-3);C(3,-3)\to A'(3,3);B'(3,1);C'(4,1)$
C. $A(-3,0);B(0,2);C(-1,3)\to A'(-9,0);B'(0,6);C'(-3,9)$
E. $A(1,-1);B(1,-5);C(5,-5)\to A'(2,-2);B'(2,-10);C'(10,-10)$