Question

∆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 .

Answer 1

`\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{√(s^2)}{√(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{√(s^2)}{√(s^2)}\implies \cfrac{1}{10}=\cfrac{√(A_1)}{√(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}`

Answer 2

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

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
`x` unit.

So, the length of longest side of ∆DEF
`= 10x`

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

`10x= 40\\ \\ x=(40)/(10)=4`

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:
`((1)/(10))^2 = (1)/(100)= 1:100`