Question

1. why are comparing side lengths within each of the two shapes (i.e., 2.4:1.7 and 0.48:0.34) and comparing corresponding side lengths between two shapes (i.e., 2.4:0.48 and 1.7:0.34) both valid methods of determining similarity?

Explanation:

Step1: Calculate ratios within shapes

For the first ratio within a shape \(2.4:1.7=\frac{2.4}{1.7}\approx1.41\). For the second ratio within a shape \(0.48:0.34=\frac{0.48}{0.34}\approx1.41\).

Step2: Calculate ratios between shapes

For the first ratio between shapes \(2.4:0.48=\frac{2.4}{0.48} = 5\). For the second ratio between shapes \(1.7:0.34=\frac{1.7}{0.34}=5\).

Step3: Use similarity criteria

If two polygons have their corresponding sides in proportion (when comparing between shapes) or the ratios of their non - corresponding sides within each shape are equal (and angles are congruent, which is assumed as we are only looking at side - length ratios for the proportion aspect here), they are similar. Here, both methods show proportionality.

Answer:

Yes, both are valid methods. When comparing side - lengths within each shape, if the ratios of non - corresponding sides within each shape are equal (and angles are congruent), the shapes can be similar. When comparing corresponding side - lengths between two shapes, if the ratios of corresponding sides are equal (and angles are congruent), the shapes are similar. In the given examples, \(\frac{2.4}{1.7}\approx\frac{0.48}{0.34}\) (within - shape comparison) and \(\frac{2.4}{0.48}=\frac{1.7}{0.34} = 5\) (between - shape comparison), so both methods can be used to determine similarity (assuming angle congruence in the geometric figures).