Calculate the surface and altitude of a triangle determined with three points

Question

The triangle is determined with three points: $A(1, -2, 8), B(0, 0, 4), C(6, 2, 0)$. How do I calculate its surface and altitude from B?

Answer 1

HINT

We can proceed as follow

• consider the line $l_{AC}$ and a generic point $P\in l_{AC}$

• consider the line $l_{BP}$ and find $\bar P$ such that $l_{B\bar P}\perp l_{AC}$ by the condition

$$v_{B\bar P}\cdot v_{AC}=0$$

• then the height is $B\bar P$ and the area is $S=\frac12 AC\cdot B\bar P$

As an alternative

• find the lenght $AB$, $BC$, $AC$ and the area by Heron's formula
• from $S=\frac12 AC \cdot H_B$ find $H_B$