2006 AMC 10B Problems/Problem 20
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Problem
In rectangle , we have
,
,
, for some integer
. What is the area of rectangle
?
Solution
Solution 1
Let the slope of be
and the slope of
be
.
Since and
form a right angle:
Using the distance formula:
Therefore the area of rectangle is
Solution 2
This solution is the same as Solution 1 up to the point where we find that .
We build right triangles so we can use the Pythagorean Theorem. The triangle with hypotenuse has legs
and
, while the triangle with hypotenuse
has legs
and
. Aha! The two triangles are similar, with one triangle having side lengths
times the other!
Let . Then from our reasoning above, we have
. Finally, the area of the rectangle is
.
See Also
2006 AMC 10B (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 |