Answer:
ZM = 8
WM = 12
XZ = 7
XN = 21
LZ = 20
Step-by-step explanation:
The centroid of a triangle is the point of intersection of the midpoint of the each of the three sides of a triangle. The centroid of a triangle is located inside the triangle and it is known as the center of gravity.
The centroid theorem for a triangle states that the centroid of a triangle is located at 2/3 of the distance from the vertex of the triangle of the middle of the opposite side.
ZM = (2/3)WM (centroid theorem)
Therefore: WZ = (1/3)WM
WM = 3WZ = 3 * 4 = 12
ZM = (2/3) * WM = 2/3 * 12 = 8
ZN = (2/3)XN (centroid theorem)
XN = (3/2)ZN = 3/2 * 14 = 21
XN = 21
XZ + ZN = XN
XZ + 14 = 21
XZ = 7
LZ = (2/3)LY (centroid theorem)
Therefore: ZY = (1/3)LY
LY = 3ZY = 3 * 10 = 30
LZ = (2/3) * LY = 2/3 * 30 = 20
ZM = 8 units, WM = 12 units, XZ = 7 units, XN = 21 units and LZ = 20 units.
Properties of the centroid of a triangle,
From the triangle given in the picture,
Z is the centroid of ΔLMN.
Centroid divides the medians XN, LY and MW in the ratio of 2 : 1.
a). ZM : ZW = 2 : 1
[tex]\frac{ZM}{ZW}=\frac{2}{1}[/tex]
[tex]\frac{ZM}{4}=\frac{2}{1}[/tex]
ZM = 8 units
b). WM = ZM + ZW
= 8 + 4
= 12 units
c). ZN : XZ = 2 : 1
[tex]\frac{ZN}{XZ}=\frac{2}{1}[/tex]
[tex]\frac{14}{XZ}=\frac{2}{1}[/tex]
XZ = 7 units
d). XN = XZ + ZN
= 7 + 14
= 21 units
e). LZ : ZY = 2 : 1
[tex]\frac{LZ}{ZY}=\frac{2}{1}[/tex]
[tex]\frac{LZ}{10}= \frac{2}{1}[/tex]
LZ = 20 units
Therefore, ZM = 8 units, WM = 12 units, XZ = 7 units, XN = 21 units and LZ = 20 units.
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