Question

unit 6 test
multiple choice
identify the choice that best completes the statement or answers the question. must show work to get credit, when necessary.

1. in $\triangle lmn$, point $p$ is the centroid, and $bp = 3$. find $lp$ and $bl$.

a. $lp = 6, bl = 9$
b. $lp = 9, bl = 12$
c. $lp = 3, bl = 6$
d. $lp = 1.5, bl = 4.5$

1. in $\triangle lmn$, point $p$ is the centroid, and $an = 18$. find $ap$ and $np$.

Explanation:

Step1: Recall Centroid Properties

The centroid of a triangle divides each median into a ratio of \(2:1\), with the longer segment being closer to the vertex. So, if \(P\) is the centroid, then for median \(LB\) (assuming \(B\) is the midpoint of \(MN\)) and median \(AN\) (assuming \(A\) is the midpoint of \(LM\)), we use the centroid ratio.

Step2: Analyze \(BP\) and \(LP\)

Given \(BP = 3\). Since the centroid divides the median into \(LP:BP=2:1\), let \(LP = 2x\) and \(BP=x\). We know \(x = 3\), so \(LP=2\times3 = 6\)? Wait, no, wait. Wait, maybe I mixed up. Wait, the median from \(L\) to \(B\) (midpoint of \(MN\)): centroid \(P\) divides \(LB\) such that \(LP:PB = 2:1\). Wait, if \(BP = 3\), then \(PB\) is the shorter segment (1 part), so \(LP\) is 2 parts. So \(LP = 2\times3=6\)? But wait, no, maybe the median is \(LN\)? Wait, no, the diagram shows \(P\) as the intersection of medians. Let's re-express.

Wait, the problem says "Find \(LP\) and \(BL\)". Wait, maybe \(BL\) is the median? Wait, no, let's check the options. Option a: \(LP = 6\), \(BL = 9\); option b: \(LP=9\), \(BL=12\); option c: \(LP=3\), \(BL=6\); option d: \(LP=1.5\), \(BL=4.5\).

Wait, centroid divides the median into \(2:1\), so the entire median length is \(3\) parts (2 + 1). If \(BP\) is 1 part (since centroid is closer to the midpoint? Wait, no: centroid is located \(\frac{2}{3}\) of the distance from the vertex to the midpoint of the opposite side. So from vertex \(L\) to midpoint \(B\) of \(MN\), the median \(LB\) has length \(LP + PB\), where \(LP=\frac{2}{3}LB\) and \(PB=\frac{1}{3}LB\).

Given \(BP = 3\), so \(PB=\frac{1}{3}LB\), so \(LB = 3\times PB=3\times3 = 9\)? Wait, no, \(PB\) is \(\frac{1}{3}LB\), so \(LB = 3\times PB\). If \(PB = 3\), then \(LB=9\), and \(LP = LB - PB=9 - 3=6\). So \(LP = 6\), \(BL = 9\) (since \(BL\) is the same as \(LB\)). That matches option a.

Wait, let's verify again. Centroid divides median into \(2:1\) (vertex to centroid : centroid to midpoint). So \(LP\) (vertex \(L\) to centroid \(P\)) is \(2\) parts, \(PB\) (centroid \(P\) to midpoint \(B\)) is \(1\) part. Given \(PB = 3\) (1 part), so \(LP = 2\times3 = 6\) (2 parts). Then total median \(LB = LP + PB=6 + 3=9\). So \(LP = 6\), \(BL = 9\) (since \(BL\) is the same as \(LB\)).

Answer:

a. \(LP = 6\), \(BL = 9\)