Problem 10 · A parallel segment in a triangle
Similarity and a 1 : 3 area ratio.
Integer answer, at most 4 digitsIn triangle $ABC$ we draw a segment $DE$ parallel to $AB$ as shown in the figure, so that the area of $DCE$ is half the area of the trapezoid $ABDE$. If side $AB$ measures 45 cm, what would be the area of a square with side $DE$?
Copa Cangur · SCM
Easy
Closed answer
Reasoned solution
Key idea: $DE \parallel AB$ makes triangles $DCE$ and $ACB$ similar, and the area ratio of similar figures is the square of the side ratio.
If $[DCE] = \tfrac{1}{2}[ABDE]$, then
$$[ACB] = [DCE] + [ABDE] = [DCE] + 2[DCE] = 3\,[DCE] \;\Longrightarrow\; \frac{[DCE]}{[ACB]} = \frac{1}{3}.$$
By similarity:
$$\left(\frac{DE}{AB}\right)^{2} = \frac{1}{3} \;\Longrightarrow\; DE^{2} = \frac{AB^{2}}{3} = \frac{2025}{3} = 675.$$
The area of the square with side $DE$ is precisely $DE^{2}$.
Answer: 675