Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade Class 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 $k^2 = \frac{11}{4}$, so the area $\text{Area} = 3$.
<div class="key-concept"><strong>Key Concept:</strong> Perpendicularity of altitudes creates a constraint that combined with the distance formula determines the triangle's dimensions.</div>
<div class="trap-box"><strong>Trap:</strong> Students may forget to use both the perpendicularity condition and the distance constraint simultaneously, leading to incomplete systems of equations.</div>
Correct Answer: 3