Question

consider △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 = 78 ), ( gj = 94 ), and ( gn = 130 ).
find ( hn ), ( lj ), and ( gk ).
note that the figure is not drawn to scale.

Explanation:

Step1: Find HN

Since N is the circumcenter, it is equidistant from all vertices of the triangle. So, \( HN = GN \). Given \( GN = 130 \), thus \( HN = 130 \).

Step2: Find LJ

\( LN \) is the perpendicular bisector of \( HJ \), so \( L \) is the midpoint of \( HJ \), but we need \( LJ \) related to \( GJ \)? Wait, no, \( LN \) is perpendicular bisector of \( HJ \)? Wait, no, the perpendicular bisectors: \( LN \) is perpendicular bisector of \( HJ \)? Wait, the figure: \( L \) is on \( HJ \), \( LN \perp HJ \), so \( L \) is midpoint? Wait, no, the problem says \( GJ = 94 \), and \( LN \) is perpendicular bisector of \( HJ \)? Wait, no, maybe \( LN \) is perpendicular bisector of \( HJ \), but actually, the circumcenter: the perpendicular bisectors of the sides meet at N. So \( KN \) is perpendicular bisector of \( GJ \), \( LN \) is perpendicular bisector of \( HJ \), \( MN \) is perpendicular bisector of \( GH \). Wait, \( GJ = 94 \), \( KN \) is perpendicular bisector of \( GJ \), so \( K \) is midpoint of \( GJ \), so \( GK = KJ = \frac{GJ}{2} \). Wait, but first, \( HN \): since N is circumcenter, \( HN = GN = 130 \) (circumradius). Then \( LJ \): \( LN \) is perpendicular bisector of \( HJ \)? Wait, no, maybe \( LN \) is perpendicular bisector of \( HJ \), but actually, the length \( LN = 78 \), but we need \( LJ \). Wait, no, maybe I made a mistake. Wait, the circumcenter: the distance from N to each vertex is equal (circumradius), so \( HN = GN = 130 \). Then, \( LN \) is the perpendicular bisector of \( HJ \), so \( L \) is the midpoint? No, wait, \( GJ = 94 \), and \( KN \) is perpendicular bisector of \( GJ \), so \( K \) is midpoint of \( GJ \), so \( GK = \frac{GJ}{2} = \frac{94}{2} = 47 \). Then, for \( LJ \): Wait, \( LN \) is perpendicular bisector of \( HJ \), but we need to find \( LJ \). Wait, no, maybe \( LN \) is perpendicular bisector of \( HJ \), so \( L \) is midpoint, but we don't know \( HJ \). Wait, no, maybe the triangle is such that \( HN = GN = 130 \), \( GK = \frac{GJ}{2} = 47 \), and \( LJ \): Wait, maybe \( LN \) is perpendicular bisector of \( HJ \), but we have \( HN = 130 \), \( LN = 78 \), so in right triangle \( LNJ \), \( LN = 78 \), \( HN = 130 \), so \( LJ = \sqrt{HN^2 - LN^2} = \sqrt{130^2 - 78^2} \). Let's calculate that: \( 130^2 = 16900 \), \( 78^2 = 6084 \), so \( 16900 - 6084 = 10816 \), \( \sqrt{10816} = 104 \)? Wait, no, that can't be. Wait, maybe \( HN = GN = 130 \) (circumradius), so \( HN = 130 \). Then \( LJ \): since \( LN \) is perpendicular bisector of \( HJ \), so triangle \( LNJ \) is right-angled at \( L \), so \( LJ = \sqrt{HN^2 - LN^2} = \sqrt{130^2 - 78^2} \). Let's compute: \( 130^2 = 16900 \), \( 78^2 = 6084 \), \( 16900 - 6084 = 10816 \), \( \sqrt{10816} = 104 \)? Wait, no, 104^2 is 10816? 100^2=10000, 4^2=16, 21004=800, so (100+4)^2=10000+800+16=10816. Yes. So \( LJ = 104 \)? Wait, no, maybe I mixed up the sides. Wait, \( HN = 130 \), \( LN = 78 \), so \( LJ = \sqrt{HN^2 - LN^2} = 104 \). Then \( GK \): since \( KN \) is perpendicular bisector of \( GJ \), \( GJ = 94 \), so \( GK = \frac{GJ}{2} = 47 \).

Step3: Verify

• \( HN \): circumradius, so \( HN = GN = 130 \).
• \( LJ \): right triangle \( LNJ \), \( LN = 78 \), \( HN = 130 \), so \( LJ = \sqrt{130^2 - 78^2} = 104 \).
• \( GK \): midpoint of \( GJ \), so \( GK = \frac{94}{2} = 47 \).

Answer:

\( HN = 130 \), \( LJ = 104 \), \( GK = 47 \)