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Difference between revisions of "1957 AHSME Problems/Problem 49"

(Created page with "== Problem == The parallel sides of a trapezoid are <math>3</math> and <math>9</math>. The non-parallel sides are <math>4</math> and <math>6</math>. A line parallel to the ba...")
 
(diagram)
Line 23: Line 23:
  
 
== Solution ==
 
== Solution ==
 +
 +
<asy>
 +
 +
import geometry;
 +
 +
defaultpen(linewidth(.8pt));
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unitsize(2cm);
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point F = origin;
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point G = (2.25,0);
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point C = (2,1);
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point B = (1,1);
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point D = 3/4*F + 1/4*B;
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point E = 3/4 * G + 1/4 * C;
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point A;
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// Defining A
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pair[] a = intersectionpoints(D--B*2, E--2*C-G);
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A = a[0];
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// Trapezoid, parallel segment
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draw(F--G--C--B--cycle);
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draw(D--E);
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// Segments AB and AC
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draw(A--B);
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draw(A--C);
 +
 +
// Points w/ labels
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dot(A);
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label("A",A,N);
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dot(B);
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label("B",B,NW);
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dot(C);
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label("C",C,NE);
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dot(D);
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label("D",D,NW);
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dot(E);
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label("E",E,NE);
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dot(F);
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label("F",F,SW);
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dot(G);
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label("G",G,SE);
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// Length Labels
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label("3",midpoint(C--B),S);
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label("9",midpoint(F--G),S);
 +
</asy>
 +
 
<math>\boxed{\textbf{(C) }4:1}</math>.
 
<math>\boxed{\textbf{(C) }4:1}</math>.
  

Revision as of 13:53, 27 July 2024

Problem

The parallel sides of a trapezoid are $3$ and $9$. The non-parallel sides are $4$ and $6$. A line parallel to the bases divides the trapezoid into two trapezoids of equal perimeters. The ratio in which each of the non-parallel sides is divided is:

[asy] defaultpen(linewidth(.8pt)); unitsize(2cm); pair A = origin; pair B = (2.25,0); pair C = (2,1); pair D = (1,1); pair E = waypoint(A--D,0.25); pair F = waypoint(B--C,0.25); draw(A--B--C--D--cycle); draw(E--F); label("6",midpoint(A--D),NW); label("3",midpoint(C--D),N); label("4",midpoint(C--B),NE); label("9",midpoint(A--B),S);[/asy]

$\textbf{(A)}\ 4: 3\qquad\textbf{(B)}\ 3: 2\qquad\textbf{(C)}\ 4: 1\qquad\textbf{(D)}\ 3: 1\qquad\textbf{(E)}\ 6: 1$

Solution

[asy] import geometry; defaultpen(linewidth(.8pt)); unitsize(2cm); point F = origin; point G = (2.25,0); point C = (2,1); point B = (1,1); point D = 3/4*F + 1/4*B; point E = 3/4 * G + 1/4 * C; point A; // Defining A pair[] a = intersectionpoints(D--B*2, E--2*C-G); A = a[0]; // Trapezoid, parallel segment draw(F--G--C--B--cycle); draw(D--E); // Segments AB and AC draw(A--B); draw(A--C); // Points w/ labels dot(A); label("A",A,N); dot(B); label("B",B,NW); dot(C); label("C",C,NE); dot(D); label("D",D,NW); dot(E); label("E",E,NE); dot(F); label("F",F,SW); dot(G); label("G",G,SE); // Length Labels label("3",midpoint(C--B),S); label("9",midpoint(F--G),S); [/asy]

$\boxed{\textbf{(C) }4:1}$.

See Also

1957 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 48 		Followed by
Problem 50
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 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50
All AHSME Problems and Solutions

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