Question

P is the centroid of △JLN, LO⊥JN, ∠NJK≅∠JNM, OP=8, OJ=15. Find the perimeter of △JLN.

Answer 1

the complete question in the attached figure

we know that
The Centroid of a Triangle is the point where the three medians of the triangle intersect.
Each median divides the triangle into two smaller triangles of equal area.
and
the centroid divides each median into two segments whose lengths are in the ratio 2:1, with the longest one nearest the vertex

therefore

triangles LOJ and LON are similar
ON=OJ=15 units
then
NJ=15+15-------> NJ=30 units

Let
x--------> distance OL
so
OP=x/3
OP=8
x/3=8--------> x=8*3---------> x=24 units
OL=24 units

in the right triangle JLO
OJ=15 units
OL=24 units
JL=?
applying the Pythagorean Theorem
JL²=OJ²+OL²--------> JL²=15²+24²------> JL=28.30 units

perimeter of triangle JLN
P=JL*2+NJ-------> P=2*28.30+30------> P=86.60 units------> P=86.6 units

the answer is
the perimeter of △JLN is 86.6 units