Question

if these two shapes are similar, what is the measure of the missing length q?
2 cm
q
8 cm
4 cm
q = \square centimeters
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work it out

Explanation:

Step1: Recall similar shapes property

Similar shapes have proportional corresponding sides. Let the blue shape's sides be \( 8 \, \text{cm} \) (longer) and \( 2 \, \text{cm} \) (shorter), and the purple shape's sides be \( 4 \, \text{cm} \) (longer) and \( q \) (shorter). So, \(\frac{8}{4}=\frac{2}{q}\) (or check ratio of scaling). Wait, actually, the ratio of corresponding sides should be equal. Wait, maybe the blue's \( 2 \, \text{cm} \) corresponds to purple's \( q \), and blue's \( 8 \, \text{cm} \) corresponds to purple's \( 4 \, \text{cm} \). So the scale factor from blue to purple is \( \frac{4}{8}=\frac{1}{2} \). Then \( q = 2\times\frac{1}{2}=1 \)? Wait, no, wait: Let's set up proportion correctly. Corresponding sides: blue's side of \( 8 \, \text{cm} \) corresponds to purple's \( 4 \, \text{cm} \), and blue's \( 2 \, \text{cm} \) corresponds to purple's \( q \). So \(\frac{8}{4}=\frac{2}{q}\)? Wait, no, that would be cross - multiply: \( 8q = 4\times2\), \( 8q = 8\), \( q = 1 \). Wait, or maybe the other way: the ratio of blue to purple for the longer sides is \( 8/4 = 2 \), so the shorter side of blue (\( 2 \)) divided by scale factor \( 2 \) gives purple's shorter side \( q = 2/2 = 1 \). Yes, that makes sense.

Step2: Solve the proportion

Set up the proportion of corresponding sides. Let the sides of the first (blue) shape be \( a = 8 \, \text{cm} \), \( b = 2 \, \text{cm} \) and the second (purple) shape be \( a' = 4 \, \text{cm} \), \( b' = q \). Since the shapes are similar, \(\frac{a}{a'}=\frac{b}{b'}\). Substitute the values: \(\frac{8}{4}=\frac{2}{q}\). Cross - multiply: \( 8\times q=4\times2 \). So \( 8q = 8 \). Divide both sides by 8: \( q=\frac{8}{8}=1 \).

Answer:

\( 1 \)