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 \(AG = 2 \times GF\).

$$ 5x + 4 = 2(3x - 1) $$

Step2: Solve for \(x\)

Expand the right side:

$$ 5x + 4 = 6x - 2 $$

Subtract \(5x\) from both sides:

$$ 4 = x - 2 $$

Add 2 to both sides:

$$ x = 6 $$

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

Substitute \(x = 6\) into \(AG = 5x + 4\):

$$ AG = 5(6) + 4 = 30 + 4 = 34 $$

Substitute \(x = 6\) into \(GF = 3x - 1\):

$$ GF = 3(6) - 1 = 18 - 1 = 17 $$

Step4: Find \(AF\)

Since \(AF = AG + GF\):

$$ AF = 34 + 17 = 51 $$

Answer:

51 units