Question

1. if z is the centroid of δwxy, wr = 87, sy = 39, and yt = 48, find each measure.

a) ws =
b) wy =
c) wz =
d) zr =
e) zt =
f) yz =

Explanation:

Step1: Recall centroid property for mid - points

The centroid divides each median in a ratio of 2:1. Also, a median bisects the opposite side. Since \(SY = 39\), then \(WS=SY = 39\) because \(S\) is the mid - point of \(WY\) (by the definition of a median).

Step2: Calculate \(WY\)

Since \(S\) is the mid - point of \(WY\) and \(WS = SY=39\), then \(WY=WS + SY=39 + 39=78\).

Step3: Recall centroid property for \(WR\)

Since \(Z\) is the centroid and \(WR = 87\), and the centroid divides the median \(WR\) in the ratio \(2:1\), let \(WZ = 2x\) and \(ZR=x\), then \(WR=WZ + ZR=2x + x=3x\). Given \(WR = 87\), we have \(3x = 87\), so \(x=\frac{87}{3}=29\). Then \(WZ = 2x=58\) and \(ZR = 29\).

Step4: Recall centroid property for \(YT\)

Since \(Z\) is the centroid and \(YT = 48\), and the centroid divides the median \(YT\) in the ratio \(2:1\), let \(YZ = 2y\) and \(ZT=y\), then \(YT=YZ + ZT=2y + y=3y\). Given \(YT = 48\), we have \(3y = 48\), so \(y = 16\). Then \(ZT = 16\) and \(YZ=32\).

Answer:

a) \(WS = 39\)
b) \(WY = 78\)
c) \(WZ = 58\)
d) \(ZR = 29\)
e) \(ZT = 16\)
f) \(YZ = 32\)