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?
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 area of the triangle is . Since the grid has an area of , the fraction of the grid covered by the triangle is .
Note angle is right; thus, the area is ; thus, the fraction of the total is .
By the Shoelace Theorem, the area of .
This means the fraction of the total area is .
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.
Using Pick's Theorem, the area of the triangle is . Therefore, the triangle takes up of the grid.
Solution 6 (Heron's Formula, Not Recommended)
We can find the lengths of the sides by using the Pythagorean Theorem. Then, we apply Heron's Formula to find the area. This simplifies to Again, we simplify to get The middle two terms inside the square root multiply to , and the first and last terms inside the square root multiply to This means that the area of the triangle is The area of the grid is Thus, the answer is .
Video Solution (HOW TO THINK CRITICALLY!!!)
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