Question

determining similarity decide whether the quadrilaterals are similar. explain your reasoning.
abcd (with d to c length 7, b to c length 3.5, right angles at a, b, c, d) and fghe (with h to g length 2, f to g length 4, right angle at f, arrows indicating sides)

1. abcd and fghe

Explanation:

Step1: Recall Similarity of Rectangles

For two rectangles (quadrilaterals with all right angles) to be similar, their corresponding side ratios must be equal. Let's denote the first rectangle \(ABCD\) with length \(l_1 = 7\) and width \(w_1 = 3.5\), and the second rectangle \(FGHE\) with length \(l_2 = 4\) and width \(w_2 = 2\) (we need to check the ratio of length to width for both).

Step2: Calculate Ratios for \(ABCD\)

The ratio of length to width for \(ABCD\) is \(\frac{l_1}{w_1}=\frac{7}{3.5}\). Simplifying, \(7\div3.5 = 2\), so \(\frac{7}{3.5}=2\).

Step3: Calculate Ratios for \(FGHE\)

The ratio of length to width for \(FGHE\) is \(\frac{l_2}{w_2}=\frac{4}{2}\). Simplifying, \(4\div2 = 2\), so \(\frac{4}{2}=2\).

Step4: Check Angle and Ratio Conditions

All angles in rectangles are right angles (\(90^\circ\)), so corresponding angles are equal. Also, the ratio of corresponding sides (length to width) is equal (\(2 = 2\)) for both rectangles.

Answer:

The quadrilaterals \(ABCD\) and \(FGHE\) are similar because all corresponding angles are equal (right angles) and the ratios of corresponding sides are equal (\(\frac{7}{3.5}=\frac{4}{2} = 2\)).