Question

four rectangles are shown in the diagram. for which dilation is the ratio of the side lengths 1:2?
a d to c
b d to b
c c to b
d b to a

Explanation:

To solve this, we analyze the dilation ratios for each option by assuming grid unit lengths (e.g., let’s assume side lengths of rectangles D, C, B, A from smallest to largest).

Step 1: Analyze Option A (D to C)

Suppose D has side lengths \( x \) and \( y \), and C has side lengths \( 1.5x \) and \( 1.5y \) (ratio \( 1.5 \), not \( 2 \)).

Step 2: Analyze Option B (D to B)

If D has side lengths \( a \) and \( b \), and B has side lengths \( 2a \) and \( 2b \), the ratio of side lengths (scale factor) is \( 2 \) (since \( \frac{\text{length of B}}{\text{length of D}} = 2 \)).

Step 3: Analyze Option C (C to B)

If C has side lengths \( m \) and \( n \), and B has side lengths \( \frac{4}{3}m \) and \( \frac{4}{3}n \) (ratio \( \frac{4}{3} \), not \( 2 \)).

Step 4: Analyze Option D (B to A)

If B has side lengths \( p \) and \( q \), and A has side lengths \( 1.5p \) and \( 1.5q \) (ratio \( 1.5 \), not \( 2 \)).

Only the dilation from D to B has a ratio of side lengths \( 1:2 \) (since B is twice as large as D in both dimensions).

Answer:

B. D to B