Question

using the centroid ratio theorem
g is the centroid of triangle abc.
what is the value of x?
what is the length of segment dg? units
what is the length of segment ag? units
what is the length of segment ad? units

Explanation:

Step1: Use centroid ratio property

The centroid ratio theorem states that \(AG = 2GD\). So, \(x + 7=2(x - 15)\).

Step2: Solve the equation for \(x\)

Expand the right - hand side: \(x+7 = 2x-30\).
Subtract \(x\) from both sides: \(7=2x - x-30\).
Add 30 to both sides: \(x=37\).

Step3: Find the length of \(DG\)

Substitute \(x = 37\) into \(DG=x - 15\). So, \(DG=37-15 = 22\) units.

Step4: Find the length of \(AG\)

Substitute \(x = 37\) into \(AG=x + 7\). So, \(AG=37 + 7=44\) units.

Step5: Find the length of \(AD\)

Since \(AD=AG + GD\), and \(AG = 44\), \(GD = 22\). Then \(AD=44+22 = 66\) units.

Answer:

\(x = 37\)
\(DG = 22\) units
\(AG = 44\) units
\(AD = 66\) units