1957 AHSME Problems/Problem 49

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

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

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

1957 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 48	Followed by Problem 50
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1957 AHSME Problems/Problem 49

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