Difference between revisions of "2008 AMC 10A Problems/Problem 20"
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Revision as of 01:48, 27 January 2022
Contents
[hide]Problem
Trapezoid has bases
and
and diagonals intersecting at
Suppose that
,
, and the area of
is
What is the area of trapezoid
?
Solution
Solution 1
![[asy] pointpen = black; pathpen = black + linewidth(0.62); /* cse5 */ pen sm = fontsize(10); /* small font pen */ pair D=(0,0),C=(12,0), K=(7,16/3); /* note that K.x is arbitrary, as generator for A,B */ pair A=7*K/4-3*C/4, B=7*K/4-3*D/4; D(MP("A",A,N)--MP("B",B,N)--MP("C",C)--MP("D",D)--A--C);D(B--D);D(A--MP("K",K)--D--cycle,linewidth(0.7)); MP("9",(A+B)/2,N,sm);MP("12",(C+D)/2,sm);MP("24",(A+D)/2+(1,0),E); [/asy]](http://latex.artofproblemsolving.com/a/8/5/a8512170c7bb9c8c44a4195790890290d8352b64.png)
Since it follows that
. Thus
.
We now introduce the concept of area ratios: given two triangles that share the same height, the ratio of the areas is equal to the ratio of their bases. Since share a common altitude to
, it follows that (we let
denote the area of the triangle)
, so
. Similarly, we find
and
.
Therefore, the area of .
See also
2008 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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 | ||
All AMC 10 Problems and Solutions |
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