Question

1. in the figures shown, △def is a scale drawing of △abc. all lengths shown are in centimeters.

a. determine the scale factor to confirm that △def is a scale drawing of △abc.

b. calculate the areas of △abc and △def.

c. what factor should you apply to the area of △abc to find the area of △def? how does the area of △def compare to the area of △abc? how does this relationship compare to the scale factor?

Explanation:

Step1: Find the scale - factor

To find the scale - factor, we can compare the corresponding side lengths. Let's take the ratio of the side lengths of $\triangle DEF$ to $\triangle ABC$. For example, comparing the hypotenuses: $\text{Scale factor}=\frac{DE}{AB}=\frac{5}{10}=\frac{1}{2}$. We can check with other sides too. $\frac{EF}{BC}=\frac{8.5}{17}=\frac{1}{2}$ and $\frac{DF}{AC}=\frac{10.5}{21}=\frac{1}{2}$.

Step2: Calculate the area of $\triangle ABC$

The area of a triangle is given by the formula $A = \frac{1}{2}\times base\times height$. For $\triangle ABC$, base $AC = 21$ cm and height $h = 8$ cm. So, $A_{ABC}=\frac{1}{2}\times21\times8=84$ $cm^{2}$.

Step3: Calculate the area of $\triangle DEF$

For $\triangle DEF$, base $DF = 10.5$ cm and height $h = 4$ cm. So, $A_{DEF}=\frac{1}{2}\times10.5\times4 = 21$ $cm^{2}$.

Step4: Find the ratio of the areas

The ratio of the area of $\triangle DEF$ to the area of $\triangle ABC$ is $\frac{A_{DEF}}{A_{ABC}}=\frac{21}{84}=\frac{1}{4}$. The scale - factor $k=\frac{1}{2}$, and the ratio of the areas is $k^{2}$.

Answer:

a. The scale factor is $\frac{1}{2}$.
b. The area of $\triangle ABC$ is $84$ $cm^{2}$ and the area of $\triangle DEF$ is $21$ $cm^{2}$.
c. The factor to apply to the area of $\triangle ABC$ to find the area of $\triangle DEF$ is $\frac{1}{4}$. The area of $\triangle DEF$ is $\frac{1}{4}$ of the area of $\triangle ABC$. The ratio of the areas is the square of the scale - factor.