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(4) A polygon with area 10 square units is dilated by a scale factor of
k. Find the area of the image for each value of
k.
PART A

k=4

◻
PART B

k=1.5

◻

{:[" PART C "],[k=1]:}

◻

{:[" PART D "],[k=(1)/(3)]:}

◻

( $4$ ) A polygon with area $10$ square units is dilated by a scale factor of $k$ . Find the area of the image for each value of $k$ . $\newline$ PART A $\newline$ $k=4$ $\newline$ $\square$ $\newline$ PART B $\newline$ $k=1.5$ $\newline$ $\square$ $\newline$ $\begin{array}{l} \text { PART C } \\ k=1 \end{array}$ $\newline$ $\square$ $\newline$ $\begin{array}{l} \text { PART D } \\ k=\frac{1}{3} \end{array}$ $\newline$ $\square$

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Volume of cubes and rectangular prisms: word problems

Full solution

Q. (

4

) A polygon with area

10

square units is dilated by a scale factor of

k

. Find the area of the image for each value of

k

.

\newline

PART A

\newline

k=4

\newline

\square

\newline

PART B

\newline

k=1.5

\newline

\square

\newline

\begin{array}{l} \text { PART C } \\ k=1 \end{array}

\newline

\square

\newline

\begin{array}{l} \text { PART D } \\ k=\frac{1}{3} \end{array}

\newline

\square

Calculate Area for $k=4$ : For PART A, $k=4$ . The area of the image after dilation is the area of the original polygon multiplied by the square of the scale factor. $\newline$ So, $\text{Area}_{\text{image}} = \text{Area}_{\text{original}} \times k^2$ .
Area for $k=4$ : Calculate the area for $k=4$ : $\text{Area}_{\text{image}} = 10 \times 4^2 = 10 \times 16$ .
Calculate Area for $k=1.5$ : Area $_{\text{image}}$ for $k=4$ is $10 \times 16 = 160$ square units.
Area for $k=1.5$ : For PART B, $k=1.5$ . Again, use the formula $\text{Area}_{\text{image}} = \text{Area}_{\text{original}} \times k^2$ .
Area for $k=1$ : Calculate the area for $k=1.5$ : $\text{Area}_{\text{image}} = 10 \times 1.5^2 = 10 \times 2.25$ .
Area for $k=1$ : $\text{Area}_{\text{image}}$ for $k=1.5$ is $10 \times 2.25 = 22.5$ square units.
Calculate Area for $k=\frac{1}{3}$ : For PART C, $k=1$ . The area of the image is the same as the original because the scale factor is $1$ .
Area for $k=\frac{1}{3}$ : $\text{Area}_{\text{image}}$ for $k=1$ is $10 \times 1^2 = 10$ square units.
Area for $k=\frac{1}{3}$ : Area $_{\text{image}}$ for $k=1$ is $10 \times 1^2 = 10$ square units.For PART D, $k=\frac{1}{3}$ . Use the formula Area $_{\text{image}}$ = Area $_{\text{original}}$ $\times k^2$ .
Area for $k=\frac{1}{3}$ : Area_image for $k=1$ is $10 \times 1^2 = 10$ square units.For PART D, $k=\frac{1}{3}$ . Use the formula $\text{Area}_{\text{image}} = \text{Area}_{\text{original}} \times k^2$ .Calculate the area for $k=\frac{1}{3}$ : $\text{Area}_{\text{image}} = 10 \times \left(\frac{1}{3}\right)^2 = 10 \times \left(\frac{1}{9}\right)$ .
Area for $k=\frac{1}{3}$ : Area $_{\text{image}}$ for $k=1$ is $10 \times 1^2 = 10$ square units.For PART D, $k=\frac{1}{3}$ . Use the formula Area $_{\text{image}}$ = Area $_{\text{original}}$ $\times k^2$ .Calculate the area for $k=\frac{1}{3}$ : Area $_{\text{image}}$ = $_{\text{image}}$ $0$ .Area $_{\text{image}}$ for $k=\frac{1}{3}$ is $_{\text{image}}$ $3$ square units.

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