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>. | ||
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+ | 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>. | ||
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+ | 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>. | ||
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+ | 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
is a point interior to rectangle and such that inches, inches, and inches. Then , in inches, equals:
Solution
From point , create perpendiculars to all four sides, labeling them starting from going north and continuing clockwise. Label the length as .
We have and , leading to .
We also have and , leading to .
Thus, , or , which is option $\boxed{\textbb{(B)}$ (Error compiling LaTeX. ! File ended while scanning use of \boxed.)
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.