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∆ABC is similar to ∆DEF. The ratio of the perimeter of ∆ABC to the perimeter of ∆DEF is 1 : 10. The longest side of ∆DEF measures 40 units. The length of the longest side of ∆ABC is units. The ratio of the area of ∆ABC to the area of ∆DEF is .

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[tex]\bf \qquad \qquad \textit{ratio relations} \\\\ \begin{array}{ccccllll} &Sides&Area&Volume\\ &-----&-----&-----\\ \cfrac{\textit{similar shape}}{\textit{similar shape}}&\cfrac{s}{s}&\cfrac{s^2}{s^2}&\cfrac{s^3}{s^3} \end{array} \\\\ -----------------------------\\\\ \cfrac{\textit{similar shape}}{\textit{similar shape}}\qquad \cfrac{s}{s}=\cfrac{\sqrt{s^2}}{\sqrt{s^2}}=\cfrac{\sqrt[3]{s^3}}{\sqrt[3]{s^3}}\\\\ -------------------------------\\\\[/tex]

[tex]\bf \cfrac{\triangle ABC}{\triangle DE F}\qquad \cfrac{longest\ side}{longest\ side}\quad \cfrac{1}{10}=\cfrac{40}{s}\implies s=\cfrac{10\cdot 40}{1}\\\\ -------------------------------\\\\ \cfrac{\triangle ABC}{\triangle DE F}\qquad \cfrac{s}{s}=\cfrac{\sqrt{s^2}}{\sqrt{s^2}}\implies \cfrac{1}{10}=\cfrac{\sqrt{A_1}}{\sqrt{A_2}}\implies \cfrac{1}{10}=\sqrt{\cfrac{A_1}{A_2}} \\\\\\ \left( \cfrac{1}{10} \right)^2=\cfrac{A_1}{A_2}\cfrac{1^2}{10^2}=\cfrac{A_1}{A_2}\implies \cfrac{1}{100}=\cfrac{A_1}{A_2}[/tex]

Answer:

The length of the longest side of ∆ABC is 4 units.

The ratio of the area of ∆ABC to the area of ∆DEF is 1 : 100

Step-by-step explanation:

The ratio of the perimeter of ∆ABC to the perimeter of ∆DEF is  1 : 10

As perimeter is one dimensional measurement, that means ∆DEF is scaled from ∆ABC  with a scale factor of 10.

Suppose, the length of longest side of ∆ABC is  [tex]x[/tex] unit.

So, the length of longest side of ∆DEF [tex]= 10x[/tex]

Given that, the longest side of ∆DEF measures 40 units. So....

[tex]10x= 40\\ \\ x=\frac{40}{10}=4[/tex]

So, the length of longest side of ∆ABC is 4 units.

Now, Area is a two dimensional measurement.

So, the ratio of the area of ∆ABC to the area of ∆DEF will be:  [tex](\frac{1}{10})^2 = \frac{1}{100}= 1:100[/tex]

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