Question

g is the centroid of triangle abc. what is the length of $overline{ae}$? units

Explanation:

Step1: Recall centroid property

The centroid of a triangle divides each median in a 2:1 ratio. So, \(AG = 2GE\). Let \(AG=2x + 10\) and \(GE = 2x-1\). Then \(2x + 10=2(2x - 1)\).

Step2: Solve the equation for \(x\)

Expand the right - hand side: \(2x+10 = 4x-2\).
Subtract \(2x\) from both sides: \(10=4x - 2-2x\), which simplifies to \(10 = 2x-2\).
Add 2 to both sides: \(10 + 2=2x\), so \(12 = 2x\).
Divide both sides by 2: \(x = 6\).

Step3: Find the length of \(AE\)

\(AE=AG + GE\).
\(AG=2x + 10\) and \(GE = 2x-1\).
Substitute \(x = 6\) into the expressions: \(AG=2\times6 + 10=12 + 10 = 22\) and \(GE=2\times6-1=12 - 1 = 11\).
\(AE=22 + 11=33\).

Answer:

33