# 1964 AHSME Problems/Problem 33

## Problem $P$ is a point interior to rectangle $ABCD$ and such that $PA=3$ inches, $PD=4$ inches, and $PC=5$ inches. Then $PB$, in inches, equals: $\textbf{(A) }2\sqrt{3}\qquad\textbf{(B) }3\sqrt{2}\qquad\textbf{(C) }3\sqrt{3}\qquad\textbf{(D) }4\sqrt{2}\qquad \textbf{(E) }2$ $[asy] pair A, B, C, D, P; A = (0, 0); B = (6.5, 0); C = (6.5, 4.5); D = (0, 4.5); P = (2.5, 1.5); draw(A--B--C--D--cycle); draw(A--P); draw(C--P); draw(D--P); draw(B--P, dashed); label("A", A, SW); label("B", B, SE); label("C", C, NE); label("D", D, NW); label("P", P, S); label("3", midpoint(A--P), NW); label("4", midpoint(D--P), NE); label("5", midpoint(C--P), NW); [/asy]$

## Solution

From point $P$, create perpendiculars to all four sides, labeling them $a, b, c, d$ starting from going north and continuing clockwise. Label the length $PB$ as $x$: $[asy] unitsize(1cm); pair A, B, C, D, P, ABfoot, BCfoot, CDfoot, DAfoot; A = (0, 0); B = (6.5, 0); C = (6.5, 4.5); D = (0, 4.5); P = (2.5, 1.5); ABfoot = (2.5, 0); BCfoot = (6.5, 1.5); CDfoot = (2.5, 4.5); DAfoot = (0, 1.5); draw(A--B--C--D--cycle); draw(A--P); draw(C--P); draw(D--P); draw(B--P, dashed); draw(ABfoot--CDfoot); draw(DAfoot--BCfoot); draw(rightanglemark(P, CDfoot, D)); draw(rightanglemark(P, BCfoot, C)); draw(rightanglemark(P, ABfoot, B)); draw(rightanglemark(P, DAfoot, A)); label("A", A, SW); label("B", B, SE); label("C", C, NE); label("D", D, NW); label("P", P, 3*dir(240)); label("3", midpoint(A--P), NW); label("4", midpoint(D--P), NE); label("5", midpoint(C--P), NW); label("x", midpoint(B--P), SW); label("a", midpoint(P--CDfoot), E); label("b", midpoint(P--BCfoot), N); label("c", midpoint(P--ABfoot), E); label("d", midpoint(P--DAfoot), N); [/asy]$

We have $a^2 + b^2 = 5^2$ and $c^2 + d^2 = 3^2$, leading to $a^2 + b^2 + c^2 + d^2 = 34$.

We also have $a^2 + d^2 = 4^2$ and $b^2 + c^2 = x^2$, leading to $a^2 + b^2 + c^2 + d^2 = 16 + x^2$.

Thus, $34 = 16 + x^2$, or $x = \sqrt{18} = 3\sqrt{2}$, which is option $\boxed{\textbf{(B)}}$

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 