Question

given △abc with medians \\(\overline{al}\\), \\(\overline{bf}\\), and \\(\overline{ce}\\) intersecting at centroid g.

if ae = 12 km, bc = 19 km, and the perimeter of △abc is 58 km, determine the length of cf.

cf = \\(\square\\) km

Explanation:

Step1: Recall the property of a median

A median of a triangle divides the opposite side into two equal parts. Since \(CE\) and \(BF\) are medians, \(AE = EB\) and \(AF=FC\).
Given \(AE = 12\) km, then \(AB=AE + EB=12 + 12=24\) km.

Step2: Use the perimeter formula

The perimeter of \(\triangle ABC\) is \(P = AB+BC + AC\). We know \(P = 58\) km, \(AB = 24\) km, and \(BC = 19\) km.
Substitute into the formula: \(58=24 + 19+AC\).
Solve for \(AC\): \(AC=58-(24 + 19)=58 - 43=15\) km.

Step3: Find the length of \(CF\)

Since \(F\) is the mid - point of \(AC\) (because \(BF\) is a median), \(CF=\frac{1}{2}AC\).
Substitute \(AC = 15\) km: \(CF=\frac{1}{2}\times15 = 7.5\) km.

Answer:

\(7.5\)