Question

for items 6–7, use △def shown. m is the centroid.

1. what is el?

a 9 c 3
b 4 d 12

1. what is an expression for fj?

a 2x c \\(\frac{4}{3}x\\)
b 3x d 4x

Explanation:

Problem 6:

Step1: Recall centroid property

In a triangle, the centroid divides each median into a ratio of \(2:1\) (from vertex to centroid : centroid to midpoint). So, \(EM:ML = 2:1\), and \(EL=EM + ML\). Given \(EM = 6\), then \(ML=\frac{EM}{2}=\frac{6}{2}=3\).

Step2: Calculate \(EL\)

\(EL=EM + ML=6 + 3 = 9\).

Step1: Recall centroid property

The centroid divides the median such that the length from the vertex to centroid is \(\frac{2}{3}\) of the median, and from centroid to midpoint is \(\frac{1}{3}\) of the median. Also, \(FM\) is part of the median, and \(FJ\) is from vertex \(F\) to centroid \(M\) plus \(MJ\)? Wait, no, actually, since \(M\) is centroid, \(FM\) is from \(F\) to \(M\), and the median from \(F\) to the midpoint of \(DE\) (let's say \(J\) is on that median? Wait, the diagram: \(FM = 2x\), and we need to find \(FJ\). Wait, no, actually, the centroid divides the median into \(2:1\), so if \(FM\) is the segment from \(F\) to \(M\) (centroid), and the full median from \(F\) to the midpoint (let's say \(J\) is the midpoint? Wait, maybe the median is \(FJ\) (from \(F\) to midpoint \(J\) of \(DE\)), and \(M\) is centroid, so \(FM:MJ = 2:1\), so \(FJ=FM + MJ\). But wait, \(FM = 2x\), and \(MJ=\frac{FM}{2}=x\)? No, wait, no: the centroid is \(\frac{2}{3}\) of the median from the vertex. So if the median is \(FJ\) (from \(F\) to midpoint \(J\) of \(DE\)), then \(FM=\frac{2}{3}FJ\), so \(FJ=\frac{3}{2}FM\)? Wait, no, the problem says \(FM = 2x\), and we need to find \(FJ\). Wait, maybe the median is from \(F\) to \(J\) (midpoint of \(DE\)), and \(M\) is centroid, so \(FM:MJ = 2:1\), so \(FM = 2x\), \(MJ = x\), so \(FJ=FM + MJ=2x + x = 3x\). Alternatively, since centroid divides the median into \(2:1\), so \(FM=\frac{2}{3}FJ\), so \(FJ=\frac{3}{2}FM=\frac{3}{2}(2x)=3x\).

Step2: Conclusion

So \(FJ = 3x\).

Answer:

A. 9

Problem 7: