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Schach [20]
2 years ago
15

∆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 measure

s 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 .
Mathematics
2 answers:
nikdorinn [45]2 years ago
8 0
\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}}\\\\
-------------------------------\\\\

\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}
goblinko [34]2 years ago
6 0

Answer:

<em>The length of the longest side of ∆ABC is 4 units.</em>

<em>The ratio of the area of ∆ABC to the area of ∆DEF is 1 : 100</em>

Step-by-step explanation:

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

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

Suppose, the length of longest side of ∆ABC is  x unit.

So, <u>the length of longest side of ∆DEF</u> = 10x

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

10x= 40\\ \\ x=\frac{40}{10}=4

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

Now, <u>Area is a two dimensional measurement</u>.

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

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