In the diagram, a shape is formed using unit squares, with $B$ the midpoint of $A C$ and $D$ the midpoint of $C E$. The line which passes through $P$ and cuts the area of the shape into two pieces of equal area also passes through the point
(A) $A$
(B) $B$
(C) $C$
(D) $D$
(E) $E$

![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-074.jpg?height=441&width=328&top_left_y=1151&top_left_x=1297)
As an initial guess, let us see what happens when the line passes through $C$.

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-320.jpg?height=504&width=377&top_left_y=813&top_left_x=928)

Since each square is a unit square, then the area of rectangle $P X C Y$ is $2 \times 3=6$, and so the line through $P$ and $C$ cuts this area in half, leaving 3 square units in the bottom piece and 3 square units in the top piece.

(A fact has been used here that will be used several times in this solution: If a line passes through two diagonally opposite vertices of a rectangle, then it cuts the rectangle into two pieces of equal area (since it cuts the rectangle into two congruent triangles).)

In the bottom piece, we have not accounted for the very bottom unit square, so the total area of the bottom piece is 4 square units.

In the top piece, we have not accounted for the two rightmost unit squares, so the total area of the top piece is 5 square units.

So putting the line through $C$ does not produce two pieces of equal area, so $C$ is not the correct answer.

Also, since the area of the bottom piece is larger than the area of the top piece when the line passes through $C$, we must move the line up to make the areas equal (so neither $A$ nor $B$ can be the answer).

Should the line pass through $E$ ?

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-321.jpg?height=487&width=415&top_left_y=239&top_left_x=909)

If so, then the line splits the square $P X E Z$ (of area 4) into two pieces of area 2.

Then accounting for the remaining squares, the area of the bottom piece is $2+3=5$ and the area of the top piece is $2+2=4$.

So putting the line through $E$ does not produce two pieces of equal area, so $E$ is not the correct answer.

By elimination, the correct answer should be $D$.

(We should verify that putting the line through $D$ does indeed split the area in half.

The total area of the shape is 9 , since it is made up of 9 unit squares.

If the line goes through $D$, the top piece consists of $\triangle P X D$ and 2 unit squares.

The area of $\triangle P X D$ is $\frac{1}{2} \times 2 \times \frac{5}{2}=\frac{5}{2}$, since $P X=2$ and $X D=\frac{5}{2}$.

Thus, the area of the top piece is $\frac{5}{2}+2=\frac{9}{2}$, which is exactly half of the total area, as required.)

ANSWER: (D)