2000 AMC 10 Problems/Problem 16
Problem
The diagram shows lattice points, each one unit from its nearest neighbors. Segment
meets segment
at
. Find the length of segment
.
Solution
Solution 1
Let be the line containing
and
and let
be the line containing
and
. If we set the bottom left point at
, then
,
,
, and
.
The line is given by the equation
. The
-intercept is
, so
. We are given two points on
, hence we can compute the slope,
to be
, so
is the line
Similarly, is given by
. The slope in this case is
, so
. Plugging in the point
gives us
, so
is the line
.
At , the intersection point, both of the equations must be true, so
We have the coordinates of and
, so we can use the distance formula here:
which is answer choice
Solution 2
Draw the perpendiculars from and
to
, respectively. As it turns out,
. Let
be the point on
for which
.
, and
, so by AA similarity,
By the Pythagorean Theorem, we have ,
, and
. Let
, so
, then
This is answer choice
Also, you could extend CD to the end of the box and create two similar triangles. Then use ratios and find that the distance is 5/9 of the diagonal of the big grid. Thus, the answer is B.
See Also
2000 AMC 10 (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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