Problem 139

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Problem 139: (2005 AMC 10a, 19)

Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated 45 degrees, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point $B$ from the line on which the bases of the original squares were placed?

$\textbf{(A)}\ 1\qquad$ $\textbf{(B)}\ \sqrt{2}\qquad$ $\textbf{(C)}\ \frac{3}{2}\qquad$ $\textbf{(D)}\ \sqrt{2}+\frac{1}{2}\qquad$ $\textbf{(E)}\ 2$

Solution

Exercise 139: Solution

Two $2$-inch squares are placed with their bases on the same line, $1$ inch apart. A third $2$-inch square is centered above the gap and rotated $45^\circ$, then lowered until it touches both squares, as shown. How many inches is the point $B$ from the line?

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