Question

which of the following is an example of two quadrilaterals that have all pairs of corresponding sides proportional but are not similar?
6 of 7 question
a square with an area of 8 square feet and a square with an area of 27 square feet
a parallelogram with adjacent sides 2.5 and 7.5 inches and an angle of 35°, and a parallelogram with adjacent sides of 3 and 9 inches and an angle of 55°
a rhombus with an area of 14 square centimeters and a 45° angle, and a rhombus with an area of 5 square centimeters and a 135° angle
a rectangle with adjacent sides 3 and 8 inches, and a rectangle with adjacent sides 1.5 and 4 inches

Explanation:

For two polygons to be similar, all corresponding sides must be proportional and all corresponding angles must be equal.

• Option A: All angles in squares are \(90^{\circ}\). If the ratio of areas of two squares is \(\frac{8}{27}\), the ratio of their side - lengths is \(\sqrt{\frac{8}{27}}=\frac{2\sqrt{2}}{3\sqrt{3}}\). But since all angles are equal (\(90^{\circ}\)), if sides are proportional (by the property of squares, if \(A = s^{2}\), \(s=\sqrt{A}\), and \(\frac{s_1}{s_2}=\sqrt{\frac{A_1}{A_2}}\)), they are similar.
• Option B: In parallelograms, the ratio of adjacent sides: \(\frac{2.5}{3}=\frac{7.5}{9}=\frac{5}{6}\). But the angles are \(35^{\circ}\) and \(55^{\circ}\) (not equal). For similar parallelograms, corresponding angles must be equal.
• Option C: In rhombuses, adjacent angles are supplementary. If one rhombus has a \(45^{\circ}\) angle, its adjacent angle is \(135^{\circ}\), and vice - versa. But for similarity, if the ratio of side - lengths (using the area formula \(A = s^{2}\sin\theta\), \(s=\sqrt{\frac{A}{\sin\theta}}\)). Let \(s_1=\sqrt{\frac{14}{\sin45^{\circ}}}\) and \(s_2=\sqrt{\frac{5}{\sin135^{\circ}}}\) (\(\sin45^{\circ}=\sin135^{\circ}=\frac{\sqrt{2}}{2}\)), \(\frac{s_1}{s_2}=\sqrt{\frac{14}{5}}\). But also, since the order of angles (if we consider corresponding angles) can be made to match (because of the supplementary property of rhombus angles), this is not the case of non - similarity when sides are proportional.
• Option D: For rectangles, all angles are \(90^{\circ}\). The ratio of adjacent sides \(\frac{3}{1.5}=\frac{8}{4} = 2\). Since all angles are equal (\(90^{\circ}\)) and sides are proportional, they are similar.

Answer:

B. A parallelogram with adjacent sides 2.5 and 7.5 inches and an angle of \(35^{\circ}\), and a parallelogram with adjacent sides of 3 and 9 inches and an angle of \(55^{\circ}\)