Question

1. figure i and figure ii are similar quadrilaterals.

figure i
diagram of figure i with sides 8 mm, 2 mm, b mm, 5 mm
figure ii
diagram of figure ii with sides a mm, 3.2 mm, 6.4 mm, 8 mm
which proportion must be true?
four multiple - choice options: \\(\frac{a}{8}=\frac{8}{2}\\), \\(\frac{3.2}{2}=\frac{a}{8}\\), \\(\frac{b}{6.4}=\frac{8}{5}\\), \\(\frac{2}{3.2}=\frac{b}{a}\\)

Explanation:

Step1: Recall similarity of figures

Similar quadrilaterals have corresponding sides in proportion. Identify corresponding sides. In Figure I: sides 2 mm, 8 mm, 5 mm, \( b \) mm. In Figure II: sides 3.2 mm, \( a \) mm, 8 mm, 6.4 mm. Corresponding sides: 2 mm ↔ 3.2 mm, 8 mm ↔ \( a \) mm, 5 mm ↔ 8 mm, \( b \) mm ↔ 6.4 mm.

Step2: Check each proportion

• For \(\frac{b}{6.4}=\frac{5}{8}\): Corresponding sides \( b \) (Figure I) and 6.4 (Figure II), 5 (Figure I) and 8 (Figure II). Proportion: \(\frac{b}{6.4}=\frac{5}{8}\) (cross - multiply: \(8b = 6.4\times5\), which is valid for similar figures as corresponding sides are proportional).

• Check other options:
• \(\frac{a}{8}=\frac{8}{2}\): Corresponding sides don't match (8 in Figure II and 8 in Figure I? No, 8 in Figure I is a different side. Incorrect.
• \(\frac{3.2}{2}=\frac{a}{8}\): Corresponding sides 3.2 (Figure II) and 2 (Figure I), \( a \) (Figure II) and 8 (Figure I)? No, 8 in Figure I is not corresponding to \( a \) here. Incorrect.
• \(\frac{2}{3.2}=\frac{b}{a}\): Sides don't correspond. Incorrect.

Answer:

\(\frac{b}{6.4}=\frac{5}{8}\) (the third option, written as \(\boldsymbol{\frac{b}{6.4}=\frac{5}{8}}\))