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rectangles 1 and 2 are similar
Step-by-step explanation:
Polygons are similar when ...
corresponding angles are congruentcorresponding sides are proportional
In a rectangle, all angles are 90°, so all angles are congruent.
The question of proportionality can be answered a couple of ways. One is to identify corresponding sides in the different figures, then see if the ratios between figures are the same.
Another way is to identify corresponding sides in the same figure, find their ratios, then see if those match with the ratios in the other figure.
For triangles and parallelograms, it can be useful to use the ratios of the sides in order from shortest to longest.
Here, the ratios are ...
rectangle 1: 5 : 6
rectangle 2: 7.5 : 9 = 15 : 18 = 5 : 6 . . . . similar to rectangle 1
rectangle 3: 12 : 15 = 4 : 5 . . . . not similar to the others
Rectangles 1 and 2 are similar.
Additional comment
Ratios are equivalent if all the numbers involved have the same factor. For rectangle 2, we wanted to reduce the ratio to the lowest integer terms. Here, we recognize that 7.5 can be cleared of a decimal by multiplying by 2, so we did ...
7.5 : 9 = (2)(7.5) : (2)(9) = 15 : 18
Now, we can see that these numbers have 3 as a common factor, so we can remove that factor.
(3)(5) : (3)(6) = 5 : 6 . . . . reduced form of the ratio 7.5 : 9.
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We can also get there by finding the greatest common divisor* of 7.5 and 9. That is 1.5, so we have ...
7.5 : 9 = (1.5)(5) : (1.5)(6) = 5 : 6
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* Euclid's algorithm for finding the GCD works with decimals or fractions as well as integers. Basically, you see if the difference is a divisor of the numbers. If it is, that is the GCD. 9 -7.5 = 1.5 is a divisor of both 7.5 and 9, so that is the GCD. (If not; the difference replaces the larger number, and the process repeats.)
