Question

1. in the figures shown, △def is a scale drawing of △abc.

a. is △def an enlargement or a reduction of △abc?

b. what is the length of the side of △def that corresponds to side ab in △abc?

c. determine the scale factor that relates △def to △abc.

1. rectangle efgh is a scale drawing of rectangle abcd. determine the scale factor, whether it is an enlargement or a reduction.

Explanation:

Step1: Compare side - lengths

Compare the side - lengths of $\triangle ABC$ and $\triangle DEF$. Sides of $\triangle ABC$ are $6$, $9$, $12$ and sides of $\triangle DEF$ are $1.5$, $2.25$, $3$. Since the side - lengths of $\triangle DEF$ are smaller, $\triangle DEF$ is a reduction of $\triangle ABC$.

Step2: Identify corresponding side

The side of $\triangle DEF$ that corresponds to side $AB$ in $\triangle ABC$ is side $DE$. The length of side $DE$ is $3$.

Step3: Calculate scale factor

The scale factor $k$ is found by dividing the length of a side of $\triangle DEF$ by the length of the corresponding side of $\triangle ABC$. Let's use side $DE$ and side $AB$. $k=\frac{DE}{AB}=\frac{3}{12}=\frac{1}{4}$.

For the rectangles:

Step4: Compare side - lengths of rectangles

Sides of rectangle $ABCD$ are $6$ and $2.25$, and sides of rectangle $EFGH$ are $4$ and $1.5$. Since the side - lengths of rectangle $EFGH$ are smaller, it is a reduction.

Step5: Calculate scale factor for rectangles

For the length - sides, if we consider the length of $AB = 6$ and $EF = 4$, the scale factor $k_1=\frac{EF}{AB}=\frac{4}{6}=\frac{2}{3}$. For the width - sides, if we consider $AD = 2.25$ and $EH = 1.5$, $k_2=\frac{EH}{AD}=\frac{1.5}{2.25}=\frac{2}{3}$. The scale factor is $\frac{2}{3}$.

Answer:

a. $\triangle DEF$ is a reduction of $\triangle ABC$.
b. The length of the side of $\triangle DEF$ that corresponds to side $AB$ in $\triangle ABC$ is $3$.
c. The scale factor that relates $\triangle DEF$ to $\triangle ABC$ is $\frac{1}{4}$.
For rectangle $EFGH$: It is a reduction and the scale factor is $\frac{2}{3}$.