Question

round to the nearest tenth if necessary
find fc.
answer attempt 4 out of 99
x =
pc =
fc =

Explanation:

Step1: Identify the property

In a triangle, if the segments from a vertex to the opposite side are medians, then the medians intersect at the centroid, and the centroid divides each median into a ratio of 2:1. Also, if \(AD\) and \(AF\) are related such that \(AD\) is a median (assuming \(D\) is the midpoint of \(AB\) and \(F\) is the midpoint of \(AC\)), but here we can see that \(AD\) and \(AF\) are parts of the median? Wait, actually, looking at the segments \(5x - 16\) (which is \(BD\)?) and \(3x + 2\) (which is \(AD\)?), no, wait, the key is that in a triangle, if \(P\) is the centroid, then \(AP = 2 \times PD\) or something, but here we have \(AD\) and \(BD\)? Wait, no, the segments \(3x + 2\) and \(5x - 16\) are probably equal because \(D\) is the midpoint? Wait, no, maybe \(AD\) and \(BD\) are equal? Wait, no, let's think again. Wait, the problem is about the centroid. The centroid divides each median into a ratio of 2:1. So if \(F\) is the midpoint of \(AC\), then \(AF = FC\)? No, \(F\) is on \(AC\), and \(PF\) is 14? Wait, no, the length from \(A\) to \(P\) and \(P\) to \(F\)? Wait, maybe the segments \(5x - 16\) and \(3x + 2\) are equal because \(D\) is the midpoint of \(AB\), so \(AD = BD\). So set \(5x - 16 = 3x + 2\).

Step2: Solve for \(x\)

Set \(5x - 16 = 3x + 2\)
Subtract \(3x\) from both sides: \(5x - 3x - 16 = 2\)
Simplify: \(2x - 16 = 2\)
Add 16 to both sides: \(2x = 2 + 16\)
\(2x = 18\)
Divide by 2: \(x = 9\)

Step3: Find \(PC\)

Now, since \(P\) is the centroid, the centroid divides the median \(FC\) (wait, no, \(PF\) is 14, and \(PC\) should be twice \(PF\) because centroid divides the median into 2:1. So \(PC = 2 \times PF\). Since \(PF = 14\), then \(PC = 2 \times 14 = 28\)

Step4: Find \(FC\)

\(FC = PF + PC = 14 + 28 = 42\)

Wait, let's check the first step again. If \(D\) is the midpoint of \(AB\), then \(AD = BD\), so \(5x - 16 = 3x + 2\). Solving:

\(5x - 3x = 2 + 16\)

\(2x = 18\)

\(x = 9\). That's correct.

Then, since \(P\) is the centroid, the median from \(B\) to \(AC\) (which is \(BF\)) is divided by \(P\) into \(BP:PF = 2:1\), but wait, no, the median from \(A\) to \(BC\) would be \(AE\), and from \(C\) to \(AB\) would be \(CD\). Wait, maybe \(F\) is the midpoint of \(AC\), so \(AF = FC\), and \(P\) is the centroid, so \(AP = 2 \times PE\), but no, the length \(PF\) is 14, so \(PC\) should be twice \(PF\) because centroid divides the median into 2:1. So \(PC = 2 \times 14 = 28\), and \(FC = PF + PC = 14 + 28 = 42\).

Answer:

\(x = 9\)
\(PC = 28\)
\(FC = 42\)