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Question

# A pediatrician's office has two doors that are similar rectangles: one for adults and one for children. The larger door has a base of 4 ft and a height of 6.5 ft. The smaller door has a base of 3 ft. Which of the following best represents the length of a diagonal on the smaller door? C. 5.1 ft, D. 5.7 ft.

Solution

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Using the ratio $\frac{\text{dimension of smaller}}{\text{dimension of larger}}$, we solve for the height of the smaller door $x$ by writing the proportion:

$\dfrac{x}{6.5}=\dfrac{3}{4}$

$x=6.5\cdot \dfrac{3}{4}$

$x=4.875\text{ ft}$

The diagonal is the hypotenuse of the right triangle formed by the rectangle where the legs are the base and height. So, by Pythagorean Theorem, we can solve for the smaller door's diagonal, $d$, as:

$d^2=3^2+4.875^2$

$d=\sqrt{3^2+4.875^2}$

$d\approx 5.7\text{ ft}$

So, the correct answer is choice $\textbf{\color{#c34632}D.}$

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