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Difference between revisions of "1993 AHSME Problems/Problem 13"

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== Solution ==
 
== Solution ==
 
Assume one of the segments bisected by the inscribed square has length <math>x</math>. Thus, the alternate segment has length <math>7-x</math>. Applying Pythagorean's Theorem, <math>x^2+(x-7)^2=5^2</math>. Simplifying, <math>(x-3)(x-4)=0</math>, so <math>x=3</math> or <math>x=4</math> (it does not matter, as rotations produce the same figure). The longest line that can be made forms a right triangle with legs  
 
Assume one of the segments bisected by the inscribed square has length <math>x</math>. Thus, the alternate segment has length <math>7-x</math>. Applying Pythagorean's Theorem, <math>x^2+(x-7)^2=5^2</math>. Simplifying, <math>(x-3)(x-4)=0</math>, so <math>x=3</math> or <math>x=4</math> (it does not matter, as rotations produce the same figure). The longest line that can be made forms a right triangle with legs  
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of <math>4</math> and <math>7</math>. <math>\sqrt{4^2+7^2}=\fbox{\sqrt{65}}</math>
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of <math>4</math> and <math>7</math>. <math>\sqrt{4^2+7^2}=\sqrt{65} \rightarrow \boxed{D}</math>
  
 
== See also ==
 
== See also ==

Latest revision as of 20:42, 7 October 2018

Problem

A square of perimeter 20 is inscribed in a square of perimeter 28. What is the greatest distance between a vertex of the inner square and a vertex of the outer square?

$\text{(A) } \sqrt{58}\quad \text{(B) } \frac{7\sqrt{5}}{2}\quad \text{(C) } 8\quad \text{(D) } \sqrt{65}\quad \text{(E) } 5\sqrt{3}$

Solution

Assume one of the segments bisected by the inscribed square has length $x$. Thus, the alternate segment has length $7-x$. Applying Pythagorean's Theorem, $x^2+(x-7)^2=5^2$. Simplifying, $(x-3)(x-4)=0$, so $x=3$ or $x=4$ (it does not matter, as rotations produce the same figure). The longest line that can be made forms a right triangle with legs of $4$ and $7$. $\sqrt{4^2+7^2}=\sqrt{65} \rightarrow \boxed{D}$

See also

1993 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
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
All AHSME Problems and Solutions

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