Question

consider $\triangle ghj$ in the figure below. the perpendicular bisectors of its sides are $\overline{kn}$, $\overline{ln}$, and $\overline{mn}$. they meet at a single point $n$. (in other words, $n$ is the circumcenter of $\triangle ghj$.) suppose $ln = 22$, $gj = 92$, and $gn = 122$. find $hn$, $lj$, and $gk$. note that the figure is not drawn to scale.

Explanation:

Step1: Hallar $HN$

El circuncentro es equidistante de los vértices del triángulo, así que $HN = GN$.
$HN = 122$

Step2: Hallar $LJ$

$LN$ es la mediatriz de $GJ$, así que $LJ = \frac{GJ}{2}$.
$LJ = \frac{92}{2} = 46$

Step3: Hallar $GK$

Usamos el teorema de Pitágoras en $\triangle GKN$: $GK = \sqrt{GN^2 - KN^2}$. Primero, notamos que $KN = LN = 22$.
$GK = \sqrt{122^2 - 22^2} = \sqrt{(122-22)(122+22)} = \sqrt{100 \times 144} = \sqrt{14400} = 120$

Answer:

$HN = 122$
$LJ = 46$
$GK = 120$