Question

do you know how?

1. find the length of each of the medians of

the triangle.
image of triangle with points a, b, c, e, d, f and markings: ae and eb marked with one tick, ad and dc marked with two ticks, cf and fb marked with three ticks; lengths 8, 24, 9 marked in red

Explanation:

Step1: Recall the property of the centroid of a triangle

The centroid of a triangle divides each median in a ratio of \(2:1\). Let the length of the median be \(m\), and the segment from the vertex to the centroid be \(2x\) and the segment from the centroid to the mid - point be \(x\).

Step2: Find the length of median \(AE\)

We know that the segment from the centroid to the mid - point \(BE = 8\). Let the length of the median \(AB\) (where \(E\) is the mid - point of \(AB\)) be \(m_{AB}\). Using the centroid ratio \(m_{AB}=3\times BE\). So \(m_{AB}=3\times8 = 24\)

Step3: Find the length of median \(CF\)

We know that the segment from the centroid to the mid - point \(GF = 9\). Let the length of the median \(CF\) (where \(F\) is the mid - point of \(AB\)) be \(m_{CF}\). Using the centroid ratio \(m_{CF}=3\times GF\). So \(m_{CF}=3\times9=27\)

Step4: Find the length of median \(BD\)

We know that the segment from the centroid to the mid - point \(GD = 24\div2 = 12\). Let the length of the median \(BD\) (where \(D\) is the mid - point of \(AC\)) be \(m_{BD}\). Using the centroid ratio \(m_{BD}=3\times12 = 36\)

Answer:

The lengths of the medians are \(24\), \(27\), and \(36\)