Questions: The quadrilaterals ABCD and PQRS are similar. Find the length x of RS.

Transcript text: The quadrilaterals $A B C D$ and $P Q R S$ are similar. Find the length $x$ of $\overline{R S}$.

Solution

Solution Steps

Step 1: Identify Corresponding Sides

Since quadrilaterals $ABCD$ and $PQRS$ are similar, their corresponding sides are proportional. Identify the corresponding sides:

$AB$ corresponds to $PQ$
$BC$ corresponds to $QR$
$CD$ corresponds to $RS$
$DA$ corresponds to $SP$

Step 2: Set Up Proportions

Using the given lengths, set up the proportion for the corresponding sides: \[ \frac{AB}{PQ} = \frac{BC}{QR} = \frac{CD}{RS} = \frac{DA}{SP} \]

Step 3: Calculate the Proportion

Using the lengths provided: \[ \frac{AB}{PQ} = \frac{3}{4.8} \] Simplify the fraction: \[ \frac{3}{4.8} = \frac{3}{4.8} = \frac{3 \div 1.2}{4.8 \div 1.2} = \frac{2.5}{4} \]

Step 4: Apply the Proportion to Find $RS$

Using the proportion: \[ \frac{CD}{RS} = \frac{2}{x} \] Since $\frac{3}{4.8} = \frac{2}{x}$, solve for $x$: \[ \frac{2}{x} = \frac{3}{4.8} \] Cross-multiply to solve for $x$: \[ 2 \cdot 4.8 = 3 \cdot x \] \[ 9.6 = 3x \] \[ x = \frac{9.6}{3} \] \[ x = 3.2 \]

Final Answer

The length of $RS$ is $3.2$.

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