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.
See Also
2015 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
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