## Task 2 - 230622

The figure $A B C D$ (see illustration) represents a rectangle composed of three equal-sized squares $A E H D, E F G H$, and $F B C G$. The line segment $A G$ intersects the line segment $E H$ at its midpoint $M$, and the line segment $B H$ intersects the line segment $F G$ at its midpoint $N$. The area of the rectangle $A B C D$ is 48 square units.

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0243.jpg?height=264&width=504&top_left_y=1242&top_left_x=296)

Determine

a) the area of triangle $S G H$,

b) the area of triangle $A B S$,

c) the area of quadrilateral $A S H D$!

Hint: You may use the theorem that any rectangle is divided into four equal triangles by its diagonals.
Due to $48: 3=16$, the area of each of the three squares is exactly 16 square units.

The line through $M$ and $N$ intersects the segment $A D$ at $P$ and

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0243.jpg?height=234&width=532&top_left_y=1682&top_left_x=268)
the segment $B C$ at $Q$. From the diagram, it can be seen:

Since $M$ is the midpoint of $E H$ and $N$ is the midpoint of $F G$, $M N G H$ is a rectangle that is half the size of the square $E F G H$. Its area is therefore 8 square units.

Similarly, the other two squares are each divided into two rectangles with an area of 8 square units by the line through $P$ and $Q$.

a) The diagonals $M G$ and $N H$ divide the rectangle $M N G H$ into four equal-area triangles. Each of them, and consequently the triangle $S G H$, has an area of 2 square units.

b) The area of the triangle $A B S$ is equal to the sum of the areas of the (mutually equal) triangles $A E M$ and $F B N$, the rectangle $E F N M$, and the triangle $M N S$.

The triangles $A E M$ and $F B N$ are each half the size of the rectangle $E F M N$, so their area is 4 square units each. Since $2 \cdot 4 + 8 + 2 = 18$, the area of the triangle $A B S$ is 18 square units.

c) The area of the quadrilateral $A S H D$ is equal to the sum of the areas of the triangle $A M P$, the rectangle $P M H D$, and the triangle $M S H$. Since $4 + 8 + 2 = 14$, the area of the quadrilateral $A S H D$ is 14 square units.