Answer:
The area of triangle DEF is [tex]4\ cm^{2}[/tex]
Step-by-step explanation:
Step 1
Find the scale factor
we know that
If triangle ABC is similar to triangle DEF
then
the ratio of their perimeters is equal to the scale factor
Let
x-------> perimeter of triangle ABC
y------> perimeter of triangle DEF
z------> scale factor
[tex]z=\frac{x}{y}[/tex]
[tex]x=5y[/tex] ------> equation A
Substitute equation A in the formula of the scale factor
[tex]z=\frac{5y}{y}[/tex]
[tex]z=5[/tex]
Step 2
Find the area of the smaller triangle DEF
we know that
The ratio of the area triangle ABC divided by the area of triangle DEF is equal to the scale factor squared
Let
r------->area of triangle ABC
s------> area of triangle DEF
z------> scale factor
[tex]z^{2}=\frac{r}{s}[/tex]
In this problem we have
[tex]z=5, r=100\ cm^{2}[/tex]
substitute and solve for s
[tex]5^{2}=\frac{100}{s}[/tex]
[tex]s=\frac{100}{5^{2}}[/tex]
[tex]s=4\ cm^{2}[/tex]
