# Problem Of The Week # 295 - Jan 04, 2018

Status
Not open for further replies.

#### Ackbach

##### Indicium Physicus
Staff member
Happy New Year! Here is this year's first University-level POTW:

-----

Tetrahedron $OABC$ is such that lines $OA, OB,$ and $OC$ are mutually perpendicular. Prove that triangle $ABC$ is not a right-angled triangle.

-----

#### Ackbach

##### Indicium Physicus
Staff member
Congratulations to Opalg and castor28 for their correct solutions to this week's POTW, which was Problem 63 in the MAA Challenges. castor28 's solution follows:

We can take an orthogonal coordinate system with $O$ as origin and one axis through each of $A, B,$ and $C$. The coordinates of the vertices are $A = (a, 0, 0), \; B=(0,b,0),$ and $C=(0,0,c)$.

We compute the dot product of the vectors $AB$ and $AC$:
$$AB\cdot AC = (-a,b,0)\cdot(-a,0,c) = a^2\ne 0.$$

This shows that $\angle BAC$ is not a right angle; the same argument applies to the other two angles of the triangle $ABC$.

Status
Not open for further replies.