Question

Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from $(a,0)$ to $(3,3),$ divides the entire region into two regions of equal area. What is $a?$

$\dfrac12$

$\dfrac35$

$\dfrac23$

$\dfrac34$

$\dfrac45$

Answer: C

Concepts:triangle area area decomposition coordinate geometry

Difficulty rating: 1540

Small Hint:

The whole region has area $5,$ so each part must be $\dfrac52$

Big Hint:

The lower-right part is a triangle of base $3-a$ and height $3,$ with one unit square removed

Solution:

The five unit squares have total area $5,$ so each region must have area $\dfrac52.$

The region to the lower right of the line is a right triangle with legs $3-a$ and $3,$ minus the one unit square it does not cover. Setting its area to $\dfrac52$ gives $\dfrac{3(3-a)}{2}-1=\dfrac52,$ so $3(3-a)=7$ and $a=\dfrac23.$

Thus, the correct answer is C.

2009 AMC 10B Problem 17

Answer: C

Difficulty rating: 1540

Small Hint:

Big Hint:

Solution:

Problem 17 in Other Years