Straight Lines
Straight Line
Allen Star Batch
Grade 11
Question:
Two vertices of a triangle are $(5, -1)$ and $(-2, 3)$. If orthocenter of the triangle is origin, then the co-ordinates of third vertex is:
$(4, 7)$
$(3, 7)$
$(-4, -7)$
$(4, -7)$
Step-by-Step Solution
Key Concept: Relationships between slopes of different line segments in a configuration can be expressed algebraically and solved simultaneously.
The slope of $AH$ is $\frac{k-7}{h-4}$ and slope of $AC$ is $\frac{k+1}{h-5}$. Setting up the relationship $3|\frac{k}{4}h + 1| = 2(h-5)$ from the given geometric condition and solving yields $\frac{k}{h} = \frac{7}{4}$ and $\frac{k+1}{h-5} = \frac{2}{3}$. These constraints produce a consistent value of $h$ and $k$ satisfying both slope conditions.
Correct Answer: 3