Question

Dilate the figure using the indicated scale factor $\mathrm{k}$. What is the value of the ratio (new to original) of the perimeters? The areas? a square with vertices $(0,0),(0,4),(4,4)$ and $(4,0);\mathrm{k}=0.5$.

Answer 1

Step 1: Find the coordinates after dilation.

Given, coordinates $\mathrm{A}(0,0),B(0,4),C(4,4)$ and $D(4,0)$ and scale factor, $\mathrm{k}=0.5$

Since, we know that after dilation the coordinates changes as:

$\mathrm{P}(\mathrm{x},\mathrm{y})=\mathrm{P}\text{'}(\mathrm{x}\times \mathrm{a},\mathrm{y}\times \mathrm{a})$

Using the above criteria of dilation, new coordinates will be

$$\begin{gathered} \mathrm{A}(0,0)=\mathrm{A}\text{'}(0\times 0.5,0\times 0.5)=\mathrm{A}\text{'}(0,0) \\ \mathrm{B}(0,4)=\mathrm{B}\text{'}(0\times 0.5,4\times 0.5)=\mathrm{B}\text{'}(0,2) \\ \mathrm{C}(4,4)=\mathrm{C}\text{'}(4\times 0.5,4\times 0.5)=\mathrm{C}\text{'}(2,2) \\ \mathrm{D}(4,0)=\mathrm{D}\text{'}(4\times 0.5,0\times 0.5)=\mathrm{D}\text{'}(2,0) \end{gathered}$$

Step 2: Find the length of sides of square.

After dilation, coordinates will be $\mathrm{A}\text{'}(0,0),\mathrm{B}\text{'}(0,2),\mathrm{C}\text{'}(2,2),\mathrm{C}\text{'}(2,0)$.

Since, we know that for length of side PQ having coordinates $\mathrm{P}({\mathrm{x}}_1,{\mathrm{y}}_1)$ and $\mathrm{Q}({\mathrm{x}}_2,{\mathrm{y}}_2)$.

So, PQ = $\sqrt{{({\mathrm{x}}_2-{\mathrm{x}}_1)}^2+{({\mathrm{y}}_2-{\mathrm{y}}_1)}^2}$

So, length for line AB is

$$\begin{gathered} \Rightarrow \mathrm{AB}=\sqrt{{(0-0)}^2+{(4-0)}^2} \\ \Rightarrow \mathrm{AB}=4 \end{gathered}$$

And, length for line A'B' is

$$\begin{gathered} \Rightarrow \mathrm{A}\text{'}\mathrm{B}\text{'}=\sqrt{{(0-0)}^2+{(2-0)}^2} \\ \Rightarrow \mathrm{A}\text{'}\mathrm{B}\text{'}=2 \end{gathered}$$

Step 3: Find the ratio of perimeters of (new to original) square.

Since, we know that when two figures are similar then,

$\Rightarrow \frac{Perimetersofnewfigure}{Perimeteroforiginalfigure}=\frac{Sidelengthofnewfigure}{Sidelengthoforiginalfigure}$

So, the ratio of perimeter of new to original figure will be

$$\begin{gathered} \Rightarrow \frac{Perimetersofnewfigure}{Perimeteroforiginalfigure}=\frac{Sidelengthofnewfigure}{Sidelengthoforiginalfigure} \\ \Rightarrow \frac{Perimetersofnewfigure}{Perimeteroforiginalfigure}=\frac{2}{4}=\frac{1}{2} \end{gathered}$$

Step 4: Find the ratio of area of (new to original) square.

Since, we know that when two figures are similar then,

$\Rightarrow \frac{Areaofnewfigure}{Areaoforiginalfigure}=\frac{{(Sidelengthofnewfigure)}^2}{{(Sidelengthoforiginalfigure)}^2}$

So, the ratio of area of new to original figure will be

$$\begin{gathered} \Rightarrow \frac{Areaofnewfigure}{Areaoforiginalfigure}=\frac{{(Sidelengthofnewfigure)}^2}{{(Sidelengthoforiginalfigure)}^2} \\ \Rightarrow \frac{Areaofnewfigure}{Areaoforiginalfigure}=\frac{2^2}{4^2}=\frac{1}{4} \end{gathered}$$

Step 5: Final answer.

Hence, the ratio of perimeter and area (new to original) is $\frac{1}{2}$ and $\frac{1}{4}$respectively.