7. Points $A(2,-4)$ and $B(3,1.5)$ are given. Point $C$ is centrally symmetric to point $B$ with respect to point $O(0,0)$, while point $D$ is axially symmetric to point $A$ with respect to the $y$-axis. Determine the area of quadrilateral $A B C D$.

No use of a pocket calculator or any reference materials is allowed.

## Ministry of Science and Education of the Republic of Croatia  Agency for Education and Upbringing  Croatian Mathematical Society

## SCHOOL/CITY COMPETITION IN MATHEMATICS
7. Points $A(2,-4)$ and $B(3,1.5)$ are given. Point $C$ is centrally symmetric to point $B$ with respect to point $O(0,0)$, while point $D$ is axially symmetric to point $A$ with respect to the $y$-axis. Determine the area of quadrilateral $A B C D$.

## Solution.

Point $C$ has coordinates $C(-3,-1.5)$.

$1 \text{POINT}$

Point $D$ has coordinates $D(-2,-4)$.

$1 \text{POINT}$

Sketch of the quadrilateral:

$1 \text{POINT}$

![](https://cdn.mathpix.com/cropped/2024_05_30_1ae5e90785d026149547g-27.jpg?height=642&width=825&top_left_y=707&top_left_x=227)

We inscribe a rectangle $E F B G$ around the quadrilateral (sketch):

![](https://cdn.mathpix.com/cropped/2024_05_30_1ae5e90785d026149547g-27.jpg?height=640&width=806&top_left_y=1433&top_left_x=228)

Then, $P_{A B C D}=P_{E F B G}-P_{A F B}-P_{B G C}-P_{C E D}$.

The lengths of the sides of rectangle $E F B G$ are $a=6$ and $b=5.5$.

$P_{E F B G}=a \cdot b=6 \cdot 5.5=33$

The lengths of the perpendicular sides of triangle $A F B$ are $a=1$ and $b=5.5$.

$P_{A F B}=a \cdot b: 2=1 \cdot 5.5: 2=2.75$

The lengths of the perpendicular sides of triangle $B G C$ are $a=6$ and $b=3$.

$P_{A F B}=a \cdot b: 2=6 \cdot 3: 2=9$

The lengths of the perpendicular sides of triangle $C E D$ are $a=1$ and $b=2.5$.

$$
P_{A F B}=a \cdot b: 2=1 \cdot 2.5: 2=1.25
$$

Then,

$P_{A B C D}=33-2.75-9-1.25=33-13=20$.

The area of quadrilateral $A B C D$ is 20.

## SCHOOL/CITY COMPETITION IN MATHEMATICS