The given information is we have the following triangle:
PART B.
If we dilate triangle ABC by a scale factor of 2, the measure of the sides of the new triangle is twice the measure of the sides of triangle ABC.
So, the area of triangle ABC is given by the formula:
[tex]A_1=\frac{b\times h}{2}=10.8[/tex]
Where b is the base and h is the height of the triangle. So, the area of the triangle after the dilation will be:
[tex]A_2=\frac{(2b)\times(2h)}{2}=\frac{4\times b\times h}{2}=4\times\frac{b\times h}{2}[/tex]
If we replace A1 in this formula, we obtain:
[tex]A_2=4\times A_1[/tex]
So, the ratio of the triangle dilated by a scale factor of 2 to the area of triangle ABC will be 4:
[tex]\frac{A_2}{A_1}=\frac{4\times A_1}{A_1}=4[/tex]
Now, if we dilate triangle ABC by a scale factor of 3, the new area will be:
[tex]\begin{gathered} A_3=\frac{3b\times3h}{2}=\frac{9\times b\times h}{2}=9\times\frac{b\times h}{2} \\ \\ By\text{ replacing A1 we have:} \\ A_3=9\times A_1 \end{gathered}[/tex]
So, the ratio of the triangle dilated by a scale factor of 3 to the area of triangle ABC will be 9:
[tex]\frac{A_3}{A_1}=\frac{9\times A_1}{A_1}=9[/tex]
PART C.
If we dilate triangle ABC by a scale factor of 1/2, the new area in terms of A1 will be:
[tex]A_{\frac{1}{2}}=\frac{\frac{1}{2}b\times\frac{1}{2}h}{2}=\frac{\frac{1}{4}\times b\times h}{2}=\frac{1}{4}\times\frac{b\times h}{2}=\frac{1}{4}\times A_1[/tex]
So, the ratio of the triangle dilated by a scale factor of 1/2 to the area of triangle ABC will be 1/4:
[tex]\frac{A_{\frac{1}{2}}}{A_1}=\frac{\frac{1}{4}A_1}{A_1}=\frac{1}{4}[/tex]
If we dilate the triangle ABC by a scale factor of 1/4, the new area will be:
[tex]A_{\frac{1}{4}}=\frac{\frac{1}{4}b\times\frac{1}{4}h}{2}=\frac{\frac{1}{16}b\times h}{2}=\frac{1}{16}\times\frac{b\times h}{2}=\frac{1}{16}A_1[/tex]
So, the ratio of the triangle dilated by a scale factor of 1/4 to the area of triangle ABC will be 1/16:
[tex]\frac{A_{\frac{1}{4}}}{A_1}=\frac{\frac{1}{16}A_1}{A_1}=\frac{1}{16}[/tex]
PART D.
Based on these observations, we can conclude the ratio of the areas of these pairs of similar triangles is always equal to the square of the scale factor.
In the first case, the scale factor was 2, and the ratio was 2^2=4
In the second case, the scale factor was 3, and the ratio of areas was 3^2=9.
In the third case, the scale factor was 1/2, and the ratio was (1/2)^2=1/4.
And finally, when the scale factor was 1/4, the ratio of the areas was (1/4)^2=1/16.
