Question

An equilateral triangle and a square have the same perimeter. What is the ratio of the length of a side of the triangle to the length of a side of the square?

Answer 1

See explanation.

Explanation:

Let the sides be:

`a` - the side of the square,

`b` - the side of the triange.

The perimeters of the figures are equal, which leads to:

`4a=3b`

If we divide both sides by `3a` we get the required ratio:

`b/a=4/3`

Answer 2

`s_e/s_s=4/3`

Explanation:

`"Perimeter of equilateral triangle"=3s_e`

`"Perimeter of a square"=4s_s`

`3s_e=4s_s`

`s_e/s_s=4/3`

Answer 3

`"Triangle's side ":" Square's side"`
`color(white)("dddddd")4color(white)("dddd.d"):color(white)("sddd")3`

Explanation:

They both have the same perimeter.

Set the total length of the perimeter as `x`

The triangle side length is `x/3`

The square side length is `x/4`

So the ratio is `x/3:x/4`

Set `x` as one length `->1` giving

`"Triangle's side ":" Square's side"`
# color(white)("dddddd")1/3color(white)("ddddd"):color(white)("sddd")1/4#

Multiply by 1 and you do not change the value. However, 1 comes in many forms

`color(white)("ddddd")color(green)( [1/3color(red)(xx1)] color(white)("d") : [1/4color(red)(xx1)] )`

#color(white)("dddd")color(green)( [1/3color(red)(xx4/4)] color(white)("d") : color(white)("d")[1/4color(red)(xx3/3)] )#

`color(white)("ddddd")color(green)( color(white)("d")[4/12] color(white)("dd") : color(white)("dd")[3/12] )`

`"Triangle's side ":" Square's side"`
`color(white)("dddddd")4color(white)("dddd.d"):color(white)("sddd")3`