2015 AMC 8 Problems/Problem 19
A triangle with vertices as ,
, and
is plotted on a
grid. What fraction of the grid is covered by the triangle?
Solution 1
The area of is equal to half the product of its base and height. By the Pythagorean Theorem, we find its height is
, and its base is
. We multiply these and divide by
to find the of the triangle is
. Since the grid has an area of
, the fraction of the grid covered by the triangle is
.
Solution 2
Note angle is right, thus the area is
thus the fraction of the total is
Solution 3
By the Shoelace theorem, the area of .
This means the fraction of the total area is
Solution 4
The smallest rectangle that follows the grid lines and completely encloses has an area of
, where
splits the rectangle into four triangles. The area of
is therefore
. That means that
takes up
of the grid.
Solution 5
Using Pick's Theorem, the area of the triangle is . Therefore, the triangle takes up
of the grid.
Solution by bobert1
See Also
2015 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
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