Question

1. are these shapes similar? yes no

Explanation:

Step1: Check the first shape (CDEF)

The first shape (CDEF) has all sides equal to 38 mi (since \( CD = DE = EF = FC = 38 \) mi) and all angles are right angles, so it's a square (or a rectangle with length = width).

Step2: Check the second shape (STRS)

The second shape (STRS) has sides: \( ST = 45 \) mi, \( TR = 36 \) mi, \( RS = 45 \) mi, \( SS = 36 \) mi (wait, actually, looking at the labels, let's correct: the second shape has sides \( ST = 45 \) mi, \( TR = 36 \) mi, \( RS = 45 \) mi, \( SS \) is a typo, probably \( SR = 36 \) mi? Wait, no, the second shape's sides: let's list the sides properly. The second shape has sides: one pair of sides 45 mi and another pair 36 mi. So it's a rectangle (since all angles are right angles) with length 45 mi and width 36 mi.

Step3: Check the ratios of corresponding sides

For two rectangles to be similar, the ratios of corresponding sides must be equal. Let's take the first shape (square/rectangle with side 38) and the second (rectangle with sides 45 and 36).

Ratio of length of first to second: \( \frac{38}{45} \)

Ratio of width of first to second: \( \frac{38}{36} \)

Since \( \frac{38}{45}
eq \frac{38}{36} \) (because 45 ≠ 36), the ratios are not equal. Wait, wait, maybe I mixed up the corresponding sides. Wait, the first shape is a square (all sides 38), the second is a rectangle with sides 36 and 45. So the sides of the first are 38, 38, 38, 38. The second are 36, 45, 36, 45. So the ratio of 38 (from first) to 36 (from second) is \( \frac{38}{36} \), and 38 to 45 is \( \frac{38}{45} \). Since these ratios are not equal, the shapes are not similar. Wait, but wait, maybe the first shape is a square (all sides 38), the second is a rectangle with length 45 and width 36. So for similarity, the ratio of length to length and width to width should be equal. But the first has length = width = 38, the second has length 45, width 36. So \( \frac{38}{45} \) (length ratio) vs \( \frac{38}{36} \) (width ratio) – not equal. Alternatively, maybe the first is a square (38x38) and the second is a rectangle (36x45). So the sides of the first are 38, 38; the second are 36, 45. So the ratio of 38 (first side) to 36 (second side) is \( \frac{38}{36} \), and 38 to 45 is \( \frac{38}{45} \). Since these are not equal, the corresponding sides are not in proportion. Therefore, the shapes are not similar.

Wait, but maybe I made a mistake. Wait, the first shape: all sides 38, so it's a square. The second shape: sides 36 and 45, so it's a rectangle. For two rectangles to be similar, the ratio of length to width must be equal. The first has length/width = \( \frac{38}{38} = 1 \). The second has length/width = \( \frac{45}{36} = \frac{5}{4} = 1.25 \). Since 1 ≠ 1.25, the ratios are not equal. Therefore, the shapes are not similar.

Answer:

no