Question

your turn
explain whether the shapes are similar.

1. rectangle abcd with sides of 7 and 5 and rectangle mnop with sides of 21 and 15

5.

Explanation:

Step1: Check the ratio of corresponding sides for rectangle

For rectangle \(ABCD\) with sides \(a = 7\), \(b=5\) and rectangle \(MNOP\) with sides \(A = 21\), \(B = 15\).
Calculate the ratios: \(\frac{A}{a}=\frac{21}{7}=3\) and \(\frac{B}{b}=\frac{15}{5}=3\).
Since all angles in rectangles are \(90^{\circ}\), and the ratios of corresponding sides are equal.

Step2: Check the ratio of corresponding sides and angles for trapezoid

For the first trapezoid with parallel sides \(8\) and \(3\), non - parallel sides \(6\) each. For the second trapezoid with parallel sides \(48\) and \(18\), non - parallel sides \(6\) each.
Calculate the ratios of parallel sides: \(\frac{48}{8}=6\) and \(\frac{18}{3}=6\), but the non - parallel sides ratio is \(\frac{6}{6} = 1\). Also, the angles of the two trapezoids are not all equal (the second trapezoid has angles \(46^{\circ},150^{\circ},110^{\circ},54^{\circ}\) and the first trapezoid has angles \(46^{\circ},150^{\circ},110^{\circ},54^{\circ}\), but the side - angle relationships for similarity (if we consider the definition of similar polygons: all corresponding angles are equal and all corresponding sides are in proportion) are not met because the non - parallel sides do not follow the same scale factor as the parallel sides.

Answer:

1. The rectangles are similar. Because all angles in rectangles are \(90^{\circ}\) (equal) and the ratios of corresponding sides \(\frac{21}{7}=\frac{15}{5} = 3\) (sides are in proportion).
2. The trapezoids are not similar. Although the ratios of the parallel sides \(\frac{48}{8}=\frac{18}{3}=6\), the ratio of non - parallel sides is \(\frac{6}{6}=1\) (not equal to the ratio of parallel sides), and for polygons to be similar, all corresponding sides must be in proportion and all corresponding angles must be equal. Here, the side - proportion condition is not fully met for all pairs of corresponding sides.