Question

point g is the centroid of triangle abc. ag = (5x + 4) units and gf = (3x − 1) units. what is af? 11 units 15 units 43 units 51 units

Explanation:

Step1: Recall centroid property

The centroid of a triangle divides each median into a ratio of \(2:1\), so \(AG = 2 \cdot GF\).
\[5x + 4 = 2(3x - 1)\]

Step2: Solve for \(x\)

Expand the right - hand side:
\[5x + 4 = 6x - 2\]
Subtract \(5x\) from both sides:
\[4=x - 2\]
Add 2 to both sides:
\[x = 6\]

Step3: Find \(AG\) and \(GF\)

First, find \(AG\): Substitute \(x = 6\) into \(AG=5x + 4\)
\[AG=5\times6 + 4=30 + 4 = 34\]
Then, find \(GF\): Substitute \(x = 6\) into \(GF = 3x-1\)
\[GF=3\times6-1=18 - 1=17\]

Step4: Find \(AF\)

Since \(AF=AG + GF\)
\[AF=34 + 17=51\]

Answer:

51 units