Problem K. 648. (January 2020)

K. 648. The sides of a square are divided into three equal parts. An interior point of the square is connected to one of the dividing points on each side, as shown in the figure, to form four quadrilaterals. Given the area of one quadrilateral (see figure), determine the areas of the other quadrilaterals.

(6 pont)

Deadline expired on February 10, 2020.

Sorry, the solution is available only in Hungarian. Google translation

Megoldás. Használjuk az ábra jelöléseit. Tudjuk, hogy a jobb alsó trapéz területe 24: \(\displaystyle t_4=\frac{(x+2x)\cdot x}{2}=24=\frac{3x^2}{2}\), vagyis \(\displaystyle 48=3x^2\), amiből \(\displaystyle 16=x^2\) és így \(\displaystyle x=4\).

\(\displaystyle t_1=x\cdot2x=4\cdot8=32,\)

\(\displaystyle t_2=\frac{(x+2x)\cdot2x}{2}=2\cdot\frac{(x+2x)\cdot x}{2}=2\cdot24=48,\)

\(\displaystyle t_3=(3x)^2-(t_1+t_2+t_3)=12^2-(24+32+48)=144-104=40.\)

Statistics:

168 students sent a solution.
6 points:	131 students.
5 points:	7 students.
4 points:	7 students.
3 points:	6 students.
2 points:	4 students.
1 point:	3 students.
0 point:	4 students.
Not shown because of missing birth date or parental permission:	6 solutions.

Problems in Mathematics of KöMaL, January 2020