# A triangle EFS, where EF is the base, is split into half with the line GH connecting to both legs. GH=20m FG=30m GS=24m HS=26m HE=32.5m EF=?

Question
Analytic geometry
A triangle EFS, where EF is the base, is split into half with the line GH connecting to both legs.
GH=20m
FG=30m
GS=24m
HS=26m
HE=32.5m
EF=?

2020-11-10
$$\displaystyle{S}\frac{{G}}{{S}}{F}=\frac{{24}}{{{24}+{30}}}=\frac{{24}}{{54}}=\frac{{4}}{{9}}$$
$$\displaystyle{S}\frac{{H}}{{S}}{E}=\frac{{26}}{{{26}+{32.5}}}=\frac{{26}}{{58.5}}=\frac{{4}}{{9}}$$
Since,
$$\displaystyle\triangle{G}{S}{H}{\quad\text{and}\quad}\triangle{F}{S}{E}$$ share $$\displaystyle\angle{S}$$, and $$\displaystyle{S}\frac{{G}}{{S}}{F}={S}\frac{{H}}{{S}}$$E, then $$\displaystyle\triangle{G}{S}{H}\sim\triangle{F}{S}{E}$$ by SAS. Thus:
$$\displaystyle{H}\frac{{G}}{{E}}{F}={S}\frac{{G}}{{S}}{F}$$
$$\displaystyle\frac{{20}}{{E}}{F}=\frac{{4}}{{9}}$$
$$\displaystyle{E}{F}=\frac{{{20}\times{9}}}{{4}}$$
EF=45m

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