Question

Consider in the figure below.
The perpendicular bisectors of its sides are , , and . They meet at a single point .
(In other words, is the circumcenter of .)
Suppose , , and .
Find , , and .
Note that the figure is not drawn to scale.

Answer 1

Let's solve

in ∆GNK

• GN=130
• GJ=94
• GK=GJ/2=47

Apply pythagorean theorem

$\\ \sf\longmapsto NK^2=GN^2-GK^2=130^2-47^2=16900-2209=14691$

$\\ \sf\longmapsto NK=\sqrt{14691}$

$\\ \sf\longmapsto NK=121(Approx)$

If you observe the traingle

• GN=NJ=130

$\\ \sf\longmapsto LJ^2=JN^2-LN^2$

$\\ \sf\longmapsto LJ^2=130^2-78^2$

$\\ \sf\longmapsto LJ^2=16900-6084$

$\\ \sf\longmapsto LJ^2=10816$

$\\ \sf\longmapsto LJ=\sqrt{10816}$

$\\ \sf\longmapsto LJ=104$

• LH=LJ =104

So

in ∆HNL

$\\ \sf\longmapsto HN^2=104^2+78^2$

$\\ \sf\longmapsto HN^2=10816+6084$

$\\ \sf\longmapsto HN^2=16900$

$\\ \sf\longmapsto HN=\sqrt{16900}$

$\\ \sf\longmapsto HN=130$

Done .