Difference between revisions of "1964 AHSME Problems/Problem 33"

(Solution)
 
(One intermediate revision by the same user not shown)
Line 29: Line 29:
 
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>.
 
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>
+
Thus, <math>34 = 16 + x^2</math>, or <math>x = \sqrt{18} = 3\sqrt{2}</math>, which is option <math>\boxed{\textbf{(B)}}</math>
 
 
 
 
  
 
==See Also==
 
==See Also==

Latest revision as of 01:35, 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 $\boxed{\textbf{(B)}}$

See Also

1964 AHSC (ProblemsAnswer KeyResources)
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

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

Invalid username
Login to AoPS