Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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));
 +
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>
 +
 
<math>\boxed{\textbf{(C) }4:1}</math>.
 
<math>\boxed{\textbf{(C) }4:1}</math>.
  

Revision as of 14:53, 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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1957_AHSME_Problems/Problem_49&oldid=223786"