Question

Please answer all questions ty :)

Answer 1

Scale factor:
`1.33333...=4/3`

Actual area:
`27`

Scale drawing area:
`48`

Ratio of areas:
`16/9`

Scale factor 2:
`4`

Scale factor
`1/3`:
`1/9`

Scale factor
`4/3`:
`16/9`

Observation: The ratio of the areas of the triangles is the square of the scale factor of the sides

Scale factor
`r`:
`r^2`

Explanation:

The scale factor is
`12/9=8/6=4/3=1.333333...`

The actual area is
`(6*9)/2=54/2=27`

The scale drawing area is
`(12*8)/2=12*4=48`

Ratio of areas:
`48/27=16/9`

When the scale factor of the sides was 2, then the value of the ratio of the areas was 4.

When the scale factor of the sides was
`1/3`, then the value of the ratio of the areas was
`1/9`.

When the scale factor of the sides was
`4/3`, then the value of the ratio of the areas was
`16/9`.

Observation: The ratio of the areas of the triangles is the square of the scale factor of the sides.

If the scale factor is
`r`, then the ratio of the areas is
`r^2`, based on the observation.

Extra: Proof of observation.

Let the legs of the actual triangle be
`a` and
`b`. Then the legs of the scale triangle are
`ra` and
`rb`, with
`r` being the scale factor.

The area of the actual triangle is
`ab/2`. The area of the scale triangle is
`ra*rb/2=r^2ab/2`.

The ratio of these areas is
`(r^2ab/2)/(ab/2)=r^2ab/ab=\boxed{r^2}`, as desired.