2004 AMC 10B Problems/Problem 18
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Problem
In the right triangle 
, we have 
, 
, and 
. Points 
, 
, and 
are located on 
, 
, and 
, respectively, so that 
, 
, and 
. What is the ratio of the area of 
to that of ?
Solution
First of all, note that 
, and therefore
.
Draw the height from onto 
as in the picture below:
Now consider the area of 
. Clearly the triangles 
and 
are similar, as they have all angles equal. Their ratio is 
, hence 
.
Now the area 
of can be computed as 
= 
.
Similarly we can find that 
as well.
Hence 
, and the answer is 
.
See also
| 2004 AMC 10B (Problems • Resources) | ||
| Preceded by Problem 17 | Followed by Problem 19 | |
| 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 | ||
