Question

In triangle STU, we have ST = SU = 22 and TU = 8. Let M be the midpoint of $\overline{ST}$ and N be on $\overline{TU}$ such that $\overline{SN}$ is an altitude of triangle STU. If $\overline{SN}$ and $\overline{UM}$ intersect at X, then what is SX?

Answer 1

Answer:

4√13  

Step-by-step explanation:

1. Calculate the length of SN

Your triangle (below) is a relatively tall isosceles triangle.

∆STN is a right triangle, so we can use Pythagoras theorem to calculate the length of SN.

SN² + NT² = ST²

SN² + 4² =22²

SN² + 16 = 484

SN² = 468

SN = √468 = 6√13

2. Calculate the length of SX

UM and SN are lines from an angle to the centre of the opposite side, so they are medians.

The medians of a triangle meet at a single point, X — the centroid.

Another characteristic is that the centroid divides each median into segments in a 2:1 ratio.

Thus,

SX = ⅔SN = ⅔ × 6√13 = 4√13