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Difference between revisions of "1965 AHSME Problems/Problem 35"

(deleted my unfinished diagram)
(diagram)
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== Solution ==
 
== Solution ==
  
 +
<asy>
  
 +
import geometry;
 +
 +
point M;
 +
segment l;
 +
 +
// Rectangle ABCD
 +
draw((0,sqrt(5))--(0,0)--(5,0)--(5,sqrt(5))--(0,sqrt(5)));
 +
dot((0,sqrt(5)));
 +
label("A", (0,sqrt(5)), NW);
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dot((0,0));
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label("B", (0,0), SW);
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dot((5,0));
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label("C", (5,0), SE);
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dot((5,sqrt(5)));
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label("D", (5, sqrt(5)), NE);
 +
 +
// Segment AC and point M
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M=(2.5,sqrt(5)/2);
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l=line((0,sqrt(5)),(5,0));
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draw(l);
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dot(M);
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label("M",M,W);
 +
 +
// Segments AX, CY, and XY
 +
pair[] x=intersectionpoints(perpendicular(M,l),(0,0)--(5,0));
 +
pair[] y=intersectionpoints(perpendicular(M,l),(0,sqrt(5))--(5,sqrt(5)));
 +
dot(x[0]);
 +
label("X",x[0],SW);
 +
dot(y[0]);
 +
label("Y",y[0],NE);
 +
draw((0,sqrt(5))--x[0]);
 +
draw((5,0)--y[0]);
 +
draw(x[0]--y[0]);
 +
 +
// Right Angle Markers
 +
markscalefactor=0.025;
 +
draw(rightanglemark((0,sqrt(5)),M,y[0])); // Angle AMY
 +
draw(rightanglemark((5,0),M,x[0])); // Angle CMX
 +
draw(rightanglemark((0,sqrt(5)),(0,0),(5,0))); // Angle ABC
 +
draw(rightanglemark((0,sqrt(5)), (5,sqrt(5)),(5,0))); // Angle ADC
 +
 +
// Length Labels
 +
label("$5$",(2.5,0),S);
 +
label("$w$",(0,sqrt(5)/2),W);
 +
 +
</asy>
  
 
<math>\fbox{D}</math>
 
<math>\fbox{D}</math>

Revision as of 13:42, 19 July 2024

Problem

The length of a rectangle is $5$ inches and its width is less than $4$ inches. The rectangle is folded so that two diagonally opposite vertices coincide. If the length of the crease is $\sqrt {6}$, then the width is:

$\textbf{(A)}\ \sqrt {2} \qquad \textbf{(B) }\ \sqrt {3} \qquad \textbf{(C) }\ 2 \qquad \textbf{(D) }\ \sqrt{5}\qquad \textbf{(E) }\ \sqrt{\frac{11}{2}}$

Solution

[asy] import geometry; point M; segment l; // Rectangle ABCD draw((0,sqrt(5))--(0,0)--(5,0)--(5,sqrt(5))--(0,sqrt(5))); dot((0,sqrt(5))); label("A", (0,sqrt(5)), NW); dot((0,0)); label("B", (0,0), SW); dot((5,0)); label("C", (5,0), SE); dot((5,sqrt(5))); label("D", (5, sqrt(5)), NE); // Segment AC and point M M=(2.5,sqrt(5)/2); l=line((0,sqrt(5)),(5,0)); draw(l); dot(M); label("M",M,W); // Segments AX, CY, and XY pair[] x=intersectionpoints(perpendicular(M,l),(0,0)--(5,0)); pair[] y=intersectionpoints(perpendicular(M,l),(0,sqrt(5))--(5,sqrt(5))); dot(x[0]); label("X",x[0],SW); dot(y[0]); label("Y",y[0],NE); draw((0,sqrt(5))--x[0]); draw((5,0)--y[0]); draw(x[0]--y[0]); // Right Angle Markers markscalefactor=0.025; draw(rightanglemark((0,sqrt(5)),M,y[0])); // Angle AMY draw(rightanglemark((5,0),M,x[0])); // Angle CMX draw(rightanglemark((0,sqrt(5)),(0,0),(5,0))); // Angle ABC draw(rightanglemark((0,sqrt(5)), (5,sqrt(5)),(5,0))); // Angle ADC // Length Labels label("$5$",(2.5,0),S); label("$w$",(0,sqrt(5)/2),W); [/asy]

$\fbox{D}$

See Also

1965 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 34 		Followed by
Problem 36
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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