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Difference between revisions of "1964 AHSME Problems/Problem 33"
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label("$5$",(4.5,3),dir(120));
 
label("$5$",(4.5,3),dir(120));
 
</asy>
 
</asy>
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 +
==Solution==
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 +
From point <math>P</math>, create perpendiculars to all four sides, labeling them <math>a, b, c, d</math> starting from going north and continuing clockwise.  Label the length <math>PB</math> as <math>x</math>.
 +
 +
We have <math>a^2 + b^2 = 5^2</math> and <math>c^2 + d^2 = 3^2</math>, leading to <math>a^2 + b^2 + c^2 + d^2 = 34</math>.
 +
 +
We also have <math>a^2 + d^2 = 4^2</math> and <math>b^2 + c^2 = x^2</math>, leading to <math>a^2 + b^2 + c^2 + d^2 = 16 + x^2</math>.
 +
 +
Thus, <math>34 = 16 + x^2</math>, or <math>x = \sqrt{18} = 3\sqrt{2}</math>, which is option <math>\boxed{\textbb{(B)}</math>
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 +
  
 
==See Also==
 
==See Also==

Revision as of 01:34, 25 July 2019

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] draw((0,0)--(6.5,0)--(6.5,4.5)--(0,4.5)--cycle); draw((2.5,1.5)--(0,0)); draw((2.5,1.5)--(0,4.5)); draw((2.5,1.5)--(6.5,4.5)); draw((2.5,1.5)--(6.5,0),linetype("8 8")); label("$A$",(0,0),dir(-135)); label("$B$",(6.5,0),dir(-45)); label("$C$",(6.5,4.5),dir(45)); label("$D$",(0,4.5),dir(135)); label("$P$",(2.5,1.5),dir(-90)); label("$3$",(1.25,0.75),dir(120)); label("$4$",(1.25,3),dir(35)); label("$5$",(4.5,3),dir(120)); [/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 $x$.

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 $\boxed{\textbb{(B)}$ (Error compiling LaTeX. Unknown error_msg)


See Also

1964 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 32 		Followed by
Problem 34
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40
All AHSME Problems and Solutions

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