The coordinates of three of the vertices of a parallelogram are $(0,0),(1,4)$ and $(4,1)$. What is the area of this parallelogram?
(A) 15
(B) 19
(C) 16
(D) 17
(E) 12

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-070.jpg?height=415&width=469&top_left_y=882&top_left_x=1294)
We label the three points as $O(0,0), P(1,4)$ and $Q(4,1)$.

There are three possible locations for the fourth vertex $R$ of the parallelogram - between $O$ and $P$ (in the second quadrant), between $P$ and $Q$ (in the first quadrant), and between $Q$ and $O$ (in the fourth quadrant).

In each of these cases, $\triangle O P Q$ will make up half of the parallelogram, and so the area of the parallelogram is twice the area of $\triangle O P Q$.

There are many ways to calculate the area of $\triangle O P Q$.

We proceed by "completing the rectangle" which includes the $x$-axis, the $y$-axis, the line $y=4$, and the line $x=4$.

We label the point $(0,4)$ as $S$, the point $(4,4)$ as $T$, and the point $(4,0)$ as $U$.

(Note that rectangle $O S T U$ is in fact a square, so we have "completed the square"!)

The area of $\triangle O P Q$ equals the area of rectangle $O S T U$ minus the combined areas of $\triangle O S P, \triangle P T Q$, and $\triangle Q U O$.

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-139.jpg?height=420&width=480&top_left_y=2202&top_left_x=1533)

The area of rectangle $O S T U$ is $4 \cdot 4=16$, since it is a square with side length 4 .

Consider $\triangle O S P$. It is right-angled at $S$, with $O S=4$ and $S P=1$.

Thus, its area is $\frac{1}{2}(O S)(S P)=\frac{1}{2}(4)(1)=2$.

Consider $\triangle P T Q$. It is right-angled at $T$, with $P T=T Q=3$.

Thus, its area is $\frac{1}{2}(P T)(T Q)=\frac{1}{2}(3)(3)=\frac{9}{2}$.

Consider $\triangle Q U O$. It is right-angled at $U$, with $O U=4$ and $U Q=1$.

Thus, its area is $\frac{1}{2}(O U)(U Q)=\frac{1}{2}(4)(1)=2$.

Therefore, the area of $\triangle O P Q$ is $16-2-\frac{9}{2}-2=\frac{15}{2}$.

Thus, the area of the parallelogram is $2 \cdot \frac{15}{2}=15$.

ANSWER: (A)