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=√(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=√(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=√(16900)`

`\\ \sf\longmapsto HN=130`

Done .