Let STU be a triangle with side lengths ST=22, TU=8 and SU=22. Let M be the midpoint of ST and let N be on TU such that SN is an altitude of triangle STU. If SN and UM intersect at X, then what is SX?

waterlili Aug 11, 2022

#1**+1 **

Note that because \(\triangle STU\) is isosceles, \(SN\) is also a median.

Now, draw the median \(TK\). This intersects at \(X\), and by the properties of medians, we know that \(SX = {2 \over 3}\times SN\).

\(\triangle SNU \) is a right triangle with \(SU = 22\) and \(NU = 8 \div 2 = 4\), meaning \(SN = \sqrt{22^2-4^2} = \sqrt{468} = 6 \sqrt{13}\)

So, \(SX = {2 \over 3} \times 6 \sqrt{13} = \color{brown}\boxed{4 \sqrt {13}}\)

BuilderBoi Aug 12, 2022