Question

select the correct answer. these two pentagons are similar. two units are added to each side of pentagon fghij. which statement about this situation correctly applies the concept of similarity? a. the pentagons remain similar because the same length was added to each side of fghij. b. the pentagons may not be similar because changing the lengths by adding the same number of units doesnt preserve similarity. c. the pentagons are not similar because changing the side lengths of fghij will also change the angle measures. d. the pentagons remain similar because similarity is not affected when the lengths of a figure are changed.

Explanation:

Step1: Recall the definition of similar figures

Similar figures have corresponding angles equal and corresponding sides in proportion.

Step2: Analyze each option

• Option A: Adding the same length to each side changes the ratio of side - lengths. For example, if the original side - lengths of one pentagon are \(a,b,c,d,e\) and the other pentagon has side - lengths \(ka,kb,kc,kd,ke\) (so they are similar with a scale factor \(k\)), after adding \(x\) to each side, the new ratios are \(\frac{a + x}{ka+x}

eq k\) (in general).

• Option B: Changing side - lengths by adding the same number of units changes the ratio of corresponding sides. Let the original sides be \(s_1,s_2\) and \(s'_1 = k s_1,s'_2=k s_2\) (similar). After adding \(n\) units, \(\frac{s_1 + n}{s'_1 + n}=\frac{s_1 + n}{k s_1 + n}

eq k\) (in general).

• Option C: Changing side - lengths changes the shape (since the ratio of corresponding sides is not maintained). Angles of a pentagon are related to its shape. If side - lengths are changed in a non - proportional way, angles (which are determined by the relative lengths of sides) will change.
• Option D: Similarity is based on proportional side - lengths. If we change the lengths of a figure (in a non - proportional way, but here the key is the property of similarity). The definition of similarity is about proportional side - lengths and equal angles. When we just change the lengths (not proportionally as in adding the same amount to each side), similarity is affected. But the correct reason is that for similarity, we need a scale factor (multiplicative change). Adding the same number of units to each side is an additive change, which does not preserve the ratio of corresponding sides.

Answer:

B. The pentagons may not be similar because changing the lengths by adding the same number of units doesn't preserve similarity.