8. Harold draws a trapezium with parallel top and bottom sides. The length of the top side is smaller than the length of the bottom side. The two diagonals divide the trapezium into four triangles. The area of the top triangle is denoted as $A$, the bottom one as $B$, the left one as $C$, and the right one as $D$. An example of such a trapezium is shown on the right.

![](https://cdn.mathpix.com/cropped/2024_04_17_51d50f3426d3742e694cg-2.jpg?height=268&width=531&top_left_y=1896&top_left_x=1385)

Which of the following equations always holds true, regardless of the exact trapezium Harold draws?
A) $A+C=B+D$
B) $A+D=B+C$
C) $A+B=C+D$
D) $A: B=D: C$
E) $A: C=D: B$

## B-questions
A8. E) $A: C=D: B$ The vertices of the trapezium are denoted by $P, Q, R$ and $S$, and the intersection of the diagonals by $T$, as shown in the figure. Relative to the diagonal $P R$, triangles $T R S$ and $T P S$ have the same height. Their areas, $A$ and $C$, therefore relate as $|T R|:|T P|$. Similarly, the ratio between $D$ and $B$ is also equal to $|T R|:|T P|$. We thus see that $A: C=D: B$. This holds for any trapezium that Harold can draw, so answer $\mathrm{E}$) is correct.

To see that the other answers are incorrect, we consider the specific trapezium in the figure. We see that area $B$ is much larger than the other three areas, so answers A), B) and C) can be immediately ruled out. Triangles $P R S$ and $Q R S$ have the same base and height, and thus also the same area. Therefore, $C$ and $D$ are also equal (this holds for any trapezium that Harold can draw). We thus see that $D: C$ is equal to 1, while $A: B$ is much smaller than 1. Therefore, answer D) can also be ruled out.

![](https://cdn.mathpix.com/cropped/2024_04_17_89d61e9f56efbfe4ed42g-3.jpg?height=451&width=899&top_left_y=1448&top_left_x=584)