Difference between revisions of "1965 AHSME Problems/Problem 8"
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== Solution == | == Solution == | ||
− | <math>\fbox{A}</math> | + | |
+ | <asy> | ||
+ | draw((0,0)--(18,0)); | ||
+ | dot((0,0)); | ||
+ | label("A", (-1.5,-0.5)); | ||
+ | dot((18,0)); | ||
+ | label("B",(19.5,-0.5)); | ||
+ | |||
+ | label("$18$", (9,-1.5)); | ||
+ | |||
+ | draw((0,0)--(5,10)--(18,0)); | ||
+ | dot((5,10)); | ||
+ | label("E", (6,11.5)); | ||
+ | |||
+ | draw((0.918,1.835)--(15.615,1.835)); | ||
+ | dot((0.918,1.835)); | ||
+ | label("D", (-0.65,2.25)); | ||
+ | dot((15.615,1.835)); | ||
+ | label("C", (17.25,2.25)); | ||
+ | |||
+ | </asy> | ||
+ | |||
+ | Let the given triangle be <math>\triangle ABE</math> with <math>AB=18</math>. Also, let <math>\overline{CD} \parallel \overline{AB}</math> with <math>C</math> on <math>\overline{BE}</math> and <math>D</math> on <math>\overline{AE}</math>. We know from the problem that <math>[ABCD]=\frac{1}{3}[\triangle ABE]</math>, so <math>[\triangle DCE]=\frac{2}{3}[\triangle ABE]</math>. Because <math>\overline{DC} \parallel \overline{AB}</math>, <math>\triangle DCE \sim \triangle ABE</math> by [[AA similarity]]. Because the ratio of these two triangles' areas is <math>\frac{2}{3}</math>, the ratio of their sides must be <math>\sqrt{\frac{2}{3}}=\frac{\sqrt{6}}{3}</math>. Thus, <math>CD=\frac{\sqrt{6}}{3}AB=\frac{\sqrt{6}}{3}*18=\boxed{6\sqrt{6}}</math>, which is answer choice <math>\fbox{A}</math>. | ||
==See Also== | ==See Also== |
Revision as of 10:53, 18 July 2024
Problem
One side of a given triangle is 18 inches. Inside the triangle a line segment is drawn parallel to this side forming a trapezoid whose area is one-third of that of the triangle. The length of this segment, in inches, is:
Solution
Let the given triangle be with
. Also, let
with
on
and
on
. We know from the problem that
, so
. Because
,
by AA similarity. Because the ratio of these two triangles' areas is
, the ratio of their sides must be
. Thus,
, which is answer choice
.
See Also
1965 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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 |