Problem K. 831. (November 2024)

K. 831. We have arranged four congruent rectangles according to the figure obtaining a large outer and a small inner square. The ratio of the areas of the large square and a single rectangle is \(\displaystyle 25:6\), and the area of the inner small square is \(\displaystyle 144~\mathrm{cm}^2\). Find the lengths of the sides of the rectangles.

(5 pont)

Deadline expired on December 10, 2024.

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

Megoldás. Legyen a nagy négyzet területe \(\displaystyle 25x\), a téglalapé \(\displaystyle 6x\). Ekkor a kis négyzet területe \(\displaystyle 25x - 4\cdot6x = x\), tehát \(\displaystyle x = 144 \textrm{ cm}^2\), azaz a nagy négyzet területe \(\displaystyle 25\cdot144 = 3600 \textrm{ cm}^2\). A nagy négyzet oldala ennek megfelelően \(\displaystyle 60 \) cm. Az ábrán a téglalapok rövidebb oldalát a-val jelölve \(\displaystyle a+12+a = 60\) cm, azaz \(\displaystyle a = 24\) cm. A téglalap hosszabbik oldala szintén az ábra alapján \(\displaystyle a+12 = 36\) cm.

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Problems in Mathematics of KöMaL, November 2024