NMTC Practice Part 4

Geometry Level 3

Medians of a triangle to the sides of lengths $\sqrt{3}$ and $\sqrt{2}$ are perpendicular. The third side's length is $x$ . Determine $x^2$ .

1 solution

Mathh Mathh
Aug 21, 2014

Hint:

Prove that the median to the third side of length $x$ is equal to $1.5x$ (the medians cut each other with the ratio $2:1$ and the median of any right triangle is equal to half of its hypotenuse).
Use the Apollonius' theorem (a special case of the Stewart's theorem ).

$(\sqrt{2})^2+(\sqrt{3})^2=2((1.5x)^2+(0.5x)^2)\implies x^2=\boxed{1}$

Excellent. Thanks for the help

Jayakumar Krishnan - 6 years, 9 months ago

This is also an NMTX-2005 problem- Savtik Golchaa

Sanjana Nedunchezian - 6 years, 9 months ago

What's NMTX? If you had meant NMTC, then thanks for telling.. :D didn't know that... is there someway I could, get the paper??

Satvik Golechha - 6 years, 8 months ago