Answer:
Scale factor: [tex]1.33333...=4/3[/tex]
Actual area: [tex]27[/tex]
Scale drawing area: [tex]48[/tex]
Ratio of areas: [tex]16/9[/tex]
Scale factor 2: [tex]4[/tex]
Scale factor [tex]1/3[/tex]: [tex]1/9[/tex]
Scale factor [tex]4/3[/tex]: [tex]16/9[/tex]
Observation: The ratio of the areas of the triangles is the square of the scale factor of the sides
Scale factor [tex]r[/tex]: [tex]r^2[/tex]
Step-by-step explanation:
The scale factor is [tex]12/9=8/6=4/3=1.333333...[/tex]
The actual area is [tex](6*9)/2=54/2=27[/tex]
The scale drawing area is [tex](12*8)/2=12*4=48[/tex]
Ratio of areas: [tex]48/27=16/9[/tex]
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 [tex]1/3[/tex], then the value of the ratio of the areas was [tex]1/9[/tex].
When the scale factor of the sides was [tex]4/3[/tex], then the value of the ratio of the areas was [tex]16/9[/tex].
Observation: The ratio of the areas of the triangles is the square of the scale factor of the sides.
If the scale factor is [tex]r[/tex], then the ratio of the areas is [tex]r^2[/tex], based on the observation.
Extra: Proof of observation.
Let the legs of the actual triangle be [tex]a[/tex] and [tex]b[/tex]. Then the legs of the scale triangle are [tex]ra[/tex] and [tex]rb[/tex], with [tex]r[/tex] being the scale factor.
The area of the actual triangle is [tex]ab/2[/tex]. The area of the scale triangle is [tex]ra*rb/2=r^2ab/2[/tex].
The ratio of these areas is [tex](r^2ab/2)/(ab/2)=r^2ab/ab=\boxed{r^2}[/tex], as desired.
