Problem 7 · The area of triangle DBE
Area ratios: one third from the base and one half from the midpoint.
Integer answer, at most 4 digitsIn the triangle below, $D$ is a point satisfying $AD = 2 \cdot CD$, and $E$ is the midpoint of $B$ and $C$. If the area of triangle $ABC$ is 546 cm², what is the area of triangle $DBE$?
Copa Cangur · SCM
Medium
Closed answer
Reasoned solution
Key idea: when two triangles share an altitude, the ratio of their areas equals the ratio of their bases.
$D$ lies on side $AC$ with $AD = 2\,CD$, so $CD = \tfrac{1}{3}\,AC$. Triangles $DBC$ and $ABC$ share the altitude from $B$ to line $AC$:
$$[DBC] = \frac{CD}{AC}\,[ABC] = \frac{546}{3} = 182\ \text{cm}^{2}.$$
Now, $E$ is the midpoint of $CB$: triangles $DBE$ and $DBC$ share the altitude from $D$ to line $CB$, and $BE = \tfrac{1}{2}\,BC$:
$$[DBE] = \frac{1}{2}\,[DBC] = \frac{182}{2} = 91\ \text{cm}^{2}.$$
Answer: 91