Question

A triangle with vertices at $A=(1,3),$ $B=(5,1),$ and $C=(4,4)$ is plotted on a $6\times5$ grid. What fraction of the grid is covered by the triangle?

$\dfrac{1}{6}$

$\dfrac{1}{5}$

$\dfrac{1}{4}$

$\dfrac{1}{3}$

$\dfrac{1}{2}$

Answer: A

Concepts:coordinate geometry triangle area

Difficulty rating: 1320

Video solution:

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Written solution:

The total area of the grid is $6\cdot5=30.$ To find the area of the triangle, place it inside the $4$ -by- $3$ rectangle shown below.

The rectangle has area $4\cdot3=12.$ The three corner triangles have areas $\dfrac12(4)(2)=4,$ $\dfrac12(1)(3)=\dfrac32,$ and $\dfrac12(3)(1)=\dfrac32.$ Therefore, $\begin{aligned} [\triangle ABC] &= 12-4-\dfrac32-\dfrac32\\ &=5. \end{aligned}$

Therefore, the fraction of the area is $\dfrac{5}{30} = \dfrac{1}{6} .$

Thus, the correct answer is A .

2015 AMC 8 Problem 19

Answer: A

Difficulty rating: 1320

Video solution:

Written solution:

Problem 19 in Other Years