Question

If $h$ is the length of the perpendicular drawn from a vertex of a regular tetrahedron to the opposite face and each edge is of length $s$., then $s^2$ is equal to_________

• A. $\frac{3h^2}{8}$
• B. $\frac{3h^2}{2}$
• C. $h^2$
• D. $\frac{8h^2}{3}$

Answer 1

The correct option is B $\frac{3h^2}{2}$

$\text{In the given figure}$

$EC=\frac{S}{2}$

$AE=\frac{\sqrt{3}}{2}S$

$\text{AS}{c}^{\prime }\text{is the centroid of}△ABC$

$\Rightarrow C^{\prime }E=\frac{1}{3}AE=\frac{S}{2\sqrt{3}}$

$C{C}^{\prime }=\frac{S}{\sqrt{3}}$

$\text{Now, In}△D{C}^{\prime }C$

$\text{Using Pythagoras theorem}$

${\left(D{C}^{\prime }\right)}^2={\left(C{C}^{\prime }\right)}^2+(CD{)}^2$

$h^2+\frac{S^2}{3}=S^2$

$\Rightarrow h^2=\frac{2S^2}{3}\Rightarrow S^2=\frac{3h^2}{2}$

Hence Option B is correct.