Question

Please help me with geometry!!

Answer 1

if we notice those tickmarks in the triangle, that simply means that those sides are twins, namely HL = LJ = 5 and so on, that also means that the line cutting them from the opposite vertex is a median.

so we have three medians, meeting at a certain point, namely the centroid.

a particular feature of the centroid is that it cuts all medians in a 2:1 ratio, namely GP : PL are in a 2:1 ratio, therefore

21)

`\bf \cfrac{GP}{PL}=\cfrac{2}{1}\implies \cfrac{8}{PL}=2\implies \cfrac{8}{2}=PL\implies 4=PL\\\\[-0.35em]~\dotfill\\\\GL=GP+PL\implies GL=8+4\implies \boxed{GL=12}`

22)

well, we already know from 21) :).

23)

recall, since the lines are medians, both sides on the opposite side are twins, HL = LJ = 5.

24)

if we take a look at the triangle GLJ, is a right-triangle, so we can use the pythagorean theorem on it,

`\bf \textit{using the pythagorean theorem}\\\\c^2=a^2+b^2\implies c=√(a^2+b^2)\qquad\begin{cases}c=\stackrel{hypotenuse}{GJ}\\a=\stackrel{adjacent}{\stackrel{LJ}{5}}\\b=\stackrel{opposite}{\stackrel{GL}{12}}\\\end{cases}\\\\\\GJ=√(5^2+12^2)\implies GJ=√(25+144)\implies GJ=√(169)\implies \boxed{GJ=13}`