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 241630 units. The ratio of the area of ∆ABC to the area of ∆DEF is 1 : 11 : 21 : 101 : 100.

Answer 1

Part 1) The length of the longest side of ∆ABC is 4 units

Part 2) The ratio of the area of ∆ABC to the area of ∆DEF is
`(1)/(100)`

Explanation:

Part 1) Find the length of the longest side of ∆ABC

we know that

If two figures are similar, then the ratio of its corresponding sides is proportional and this ratio is called the scale factor

The ratio of its perimeters is equal to the scale factor

Let

z ----> the scale factor

x ----> the length of the longest side of ∆ABC

y ----> the length of the longest side of ∆DEF

so

`z=(x)/(y)`

we have

`z=(1)/(10)`

`y=40\ units`

substitute

`(1)/(10)=(x)/(40)`

solve for x

`x=(40)(1)/(10)`

`x=4\ units`

therefore

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

Part 2) Find the ratio of the area of ∆ABC to the area of ∆DEF

we know that

If two figures are similar, then the ratio of its areas is equal to the scale factor squared

Let

z ----> the scale factor

x ----> the area of ∆ABC

y ----> the area of ∆DEF

`z^(2)=(x)/(y)`

we have

`z=(1)/(10)`

so

`z^2=((1)/(10))^2`

`z^2=(1)/(100)`

therefore

The ratio of the area of ∆ABC to the area of ∆DEF is
`(1)/(100)`