Question

the centroid of triangle abc is at point d, as shown below. * if cd = 16 cm, what is the length of ed? 4 cm 32 cm 24 cm 8 cm

Explanation:

Step1: Recall centroid property

The centroid of a triangle divides each median into a ratio of \(2:1\), with the longer segment being closer to the vertex. Also, \(E\) is the midpoint of \(AB\) (since \(E\), \(F\), \(G\) are midpoints as they are connected to centroid), so \(CE\) is a median? Wait, no, \(E\) is midpoint of \(AB\), \(G\) midpoint of \(BC\), \(F\) midpoint of \(AC\). So \(AE = EB\), \(BG=GC\), \(AF = FC\). The centroid \(D\) divides the median (e.g., from \(C\) to \(E\)?) Wait, no, the median from \(C\) would be to midpoint of \(AB\), which is \(E\). So \(CE\) is a median? Wait, no, \(E\) is midpoint of \(AB\), so the median from \(C\) is \(CE\)? Wait, no, median is from vertex to midpoint of opposite side. So from \(C\) to midpoint of \(AB\) (which is \(E\)), so \(CE\) is a median. Then centroid \(D\) divides \(CE\) into \(CD:DE = 2:1\).

Step2: Apply the ratio

Given \(CD = 16\) cm, and \(CD:DE = 2:1\), let \(DE = x\). Then \(\frac{CD}{DE}=\frac{2}{1}\), so \(\frac{16}{x}=\frac{2}{1}\). Cross - multiplying gives \(2x = 16\), so \(x=\frac{16}{2}=8\) cm.

Answer:

8 cm