Final answer:
The error is the student claimed that two rectangles are similar when the ratios of their corresponding sides are not equal. The sides of the rectangles are not proportional, and their scale factors are different, which proves they are not similar.
Step-by-step explanation:
The error in the student's statement that two rectangles with dimensions 10 in. by 9 in. and 5 in. by 3 in. are similar is that the ratios of the corresponding sides of the rectangles are not equal. For rectangles to be similar, the ratio of the lengths to the widths of the rectangles must be the same. In this case, the ratio of the larger rectangle is 10/9 while the ratio of the smaller rectangle is 5/3. These two ratios can be simplified to 10/9 and 5/3 respectively, and since 10/9 ≠5/3, the rectangles are not similar.
Furthermore, scale factor is an important concept when determining the similarity of figures. The scale factor from the larger to the smaller rectangle in terms of width would be 10 in. / 5 in. = 2, and in terms of length it would be 9 in. / 3 in. = 3. The scale factors are not the same, which further proves the rectangles are not similar.
The student should review the rules for determining similarity and understand that for geometric figures like rectangles to be similar, all corresponding sides must be in proportion.