Straight Lines
Straight Line
Allen Star Batch
Grade 11
Question:
In a triangle $ABC$ if $AC = 3$, $BC = 4$ and median $AD$ and $BE$ are perpendicular to each other, $\triangle$ be the area of the triangle $ABC$. Then the value of $[\triangle]$ is ______ (where $[.]$ denotes the greatest integer).
Step-by-Step Solution
Key Concept: Perpendicularity of altitudes creates a constraint that combined with the distance formula determines the triangle's dimensions.
Given area of triangle $ABC = 2k$ and $AD \perp BE$, we use the perpendicularity condition $k^2 + (h+4)(h-2) = 0$. With $AC = 3$, we get $(h-4)^2 + k^2 = 9$. Solving these simultaneously yields $k^2 = rac{11}{4}$, so the area $[A] = 3$.
Correct Answer: 3