Ratio of the Area of Similar Polygons

[Hobbes1]

Two corresponding sides of two similar polygons have lengths 3 and 7. the perimeter of the larger polygon is 91 cm. What is the perimeter of the smaller polygon? What is the ratio of their areas?

I believe I have found the perimeter of the smaller polygon (39), but I can't figure out the areas.

Thanks.

 

[MarkFL]

I will use the subscript $S$ for the smaller polygon, and $L$ for the larger...

For similar planar shapes, the ratio of any corresponding linear measures will be the same. And so the perimeter $P_S$ of the smaller polygon will be:

\(\displaystyle P_S=\frac{3}{7}P_L=\frac{3}{7}\cdot91=\frac{3\cdot7\cdot13}{7}=3\cdot13=39\)

So, you did find the correct value there. (Yes)

Because the area of a planar shape varies as the square of any of its linear measures, then we will find:

\(\displaystyle \frac{A_S}{A_L}=\left(\frac{3}{7}\right)^2=\frac{9}{49}\)