3D Geometry
Three Dimensional Geometry
star_batch_jee_advanced_2025
Grade 12
Question:
An equilateral triangle has its vertices on the axes of coordinates and area $\sqrt{3}$ square units. The coordinates of the orthocenter of the triangle are:
$\left(1, 1, 1\right)$
$\left(\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}\right)$
$\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right)$
$\left(\frac{\sqrt{2}}{3}, -\frac{\sqrt{2}}{3}, -\frac{\sqrt{2}}{3}\right)$
Step-by-Step Solution
Key Concept: The centroid of a triangle is the average of its three vertex coordinates.
For an equilateral triangle with side length 2, the vertices are located at $(\sqrt{2}, 0, 0)$, $(0, \sqrt{2}, 0)$, and $(0, 0, \sqrt{2})$ in 3D space. The centroid of this triangle is the average of the three vertices: $\left(\frac{\sqrt{2}}{3}, \frac{\sqrt{2}}{3}, \frac{\sqrt{2}}{3}\right)$.
Correct Answer: Let me work through this problem step by step.
**Given Information:**
- Equilateral triangle with vertices on coordinate axes
- Area = √3 square units
- Need to find orthocenter coordinates
**Step 1: Find the side length**
For an equilateral triangle with side length a:
$$\text{Area} = \frac{\