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Geometry and Trigonometry / Area and volume Difficulty: Medium

Rectangles $A B C D$ and $Q R S T$ shown are similar, where $A$ , $B$ , $C$ , and $D$ correspond to $Q$ , $R$ , $S$ , and $T$ , respectively. What is the length, in inches $(in)$ , of $\bar{A D}$ ?

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Explanation

Choice B is correct. It’s given that rectangles $A B C D$ and $Q R S T$ are similar, where $A$ , $B$ , $C$ , and $D$ correspond to $Q$ , $R$ , $S$ , and $T$ , respectively. It follows that $\bar{A B}$ corresponds to $\bar{Q R}$ and $\bar{A D}$ corresponds to $\bar{Q T}$ . If two rectangles are similar, then the lengths of their corresponding sides are proportional. It’s given in the figure that the length of $\bar{A B}$ is $12$ inches, the length of $\bar{Q R}$ is $5$ inches, and the length of $\bar{Q T}$ is $10$ inches. If $x$ is the length, in inches, of $\bar{A D}$ , then $StartFraction 12 Over 5 EndFraction$ is equivalent to $StartFraction x Over 10 EndFraction$ . Therefore, the value of $x$ can be found using the equation $\frac{12}{5} = \frac{x}{10}$ . Multiplying each side of this equation by $10$ yields $\frac{120}{5} = x$ , or $24 equals x$ . Therefore, the length, in inches, of $\bar{A D}$ is $24$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the length, in inches, of $\bar{Q T}$ , not $\bar{A D}$ .