Question

A 4 by 5 inch photo is enlarged by multiplying every dimension by 2 to form a similar 8 by 10 inch photo. What is the ratio of the perimeter of the smaller rectangle to that of the larger? What is the ratio of the two areas?

Answer 1

`1:2" and "1:4`

Explanation:

`"the ratios of the perimeters being linear is"`

`1:2`

`"and the ratios of the corresponding areas is the "`
`"square of the linear ratio"`

`1^2:2^2=1:4`

Answer 2

`color(blue)(1:2 and 1:4`

Explanation:

If we increase by a factor of 2, then the perimeter will also be increased by a factor of 2:

Proof:

Let a rectangle have sides `bba` and `bb(b)`

`"Perimeter"="sum of all sides"`

`P=a+a+b+b=2a+2b`

Enlarging by a scale factor of 2:

`2P=2a+2a+2b+2b=4a+4b=2(2a+2b)`

For area:

If we increase by a factor 2, then the area increases by a factor of `2^2`.

Proof:

Let sides be `bba` and `bb(b)`

`Area"=axxb=ab`

Increasing by a scale factor of 2:

`"Area"=2axx2b=4ab=4(ab)`