Let A B C D be a tetrahedron such that the edges A B, A C and A D are mutually perpendicul…

Question

Let $A B C D$ be a tetrahedron such that the edges $A B, A C$ and $A D$ are mutually perpendicular. Let the areas of the triangles $\mathrm{ABC}, \mathrm{ACD}$ and ADB be 5,6 and 7 square units respectively. Then the area (in square units) ...

Accepted Answer

$$\begin{aligned} & \operatorname{ar}(\triangle A B C)=5 \ & \frac{1}{2} \times b c=5 \ & \Rightarrow \quad b c=10 \ & \operatorname{ar}(\triangle A C D)=6 \ & \frac{1}{2} \times c d=6 \ & \Rightarrow \quad c d=12 \ & \operatorname{ar}(\triangle A B D)=7 \ & b d=14 \end{aligned}$$ $$\begin{aligned} & \operatorname{area}(\triangle B C D)=\frac{1}{2}|\overrightarrow{B C} \times \overrightarrow{B D}| \ & \overrightarrow{B C}=<-b, c, 0> \ & \overrightarrow{B D}=<-b, 0, d> \ & |\overrightarrow{B C} \times \overrightarrow{B D}|=\sqrt{c^2 d^2+b^2 d^2+b^2 c^2} \ & =\sqrt{12^2+14^2+10^2} \ & =\sqrt{440} \ & \operatorname{ar}(\triangle B C D)=\sqrt{110} \end{aligned}$$

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