Question

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

if \\(ae = 15\\) km, \\(bc = 27\\) km, and the perimeter of \\(\triangle abc\\) is 79 km, determine the length of \\(cf\\).

\\(cf = \square\\) km

Explanation:

Step1: Use the property of median

Since \( \overline{CE}\) and \( \overline{BF}\) are medians, \(AE = EB\) and \(CF=FA\). Given \(AE = 15\) km, then \(AB=AE + EB=15 + 15=30\) km.

Step2: Use the formula for perimeter

The perimeter of \(\triangle ABC\) is \(P = AB+BC + AC\). We know \(P = 79\) km, \(AB = 30\) km, and \(BC = 27\) km. So, \(AC=P-(AB + BC)=79-(30 + 27)=22\) km.

Step3: Find the length of \(CF\)

Since \(CF=\frac{1}{2}AC\) (because \(F\) is the mid - point of \(AC\)), then \(CF=\frac{1}{2}\times22 = 11\) km.

Answer:

\(11\)