Question

in the diagram, gb = 2x + 3.. what is gb?
○ 5 units
○ 10 units
○ 15 units
○ 30 units
diagram: points f, b, d, g, a, c, e with segments (fb, bd: 1 tick; fc, ce: 2 ticks; da, ae: 1 tick); fg = 5x, ga = x + 9, gb = 2x + 3

Explanation:

Step1: Identify the property of the centroid

In a triangle, the centroid divides each median into a ratio of \(2:1\). Also, the segments \(FG\) and \(GA\) are parts of a median, so \(FG = 2\cdot GA\) (since centroid is the intersection of medians and divides them in \(2:1\) ratio). So we have the equation \(5x=2(x + 9)\).

Step2: Solve for \(x\)

Expand the right - hand side: \(5x=2x + 18\)
Subtract \(2x\) from both sides: \(5x-2x=2x + 18-2x\), which gives \(3x = 18\)
Divide both sides by 3: \(x=\frac{18}{3}=6\)

Step3: Find the length of \(GB\)

We know that \(GB = 2x+3\). Substitute \(x = 6\) into the expression: \(GB=2\times6 + 3=12 + 3=15\)

Answer:

15 units