Answer:
Part 1)
a) ![AG=10\ units](https://tex.z-dn.net/?f=AG%3D10%5C%20units)
b) ![GD=5\ units](https://tex.z-dn.net/?f=GD%3D5%5C%20units)
c) ![CD=12\ units](https://tex.z-dn.net/?f=CD%3D12%5C%20units)
d) ![GE=6.5\ units](https://tex.z-dn.net/?f=GE%3D6.5%5C%20units)
e) ![GB=13\ units](https://tex.z-dn.net/?f=GB%3D13%5C%20units)
Part 2)
a) ![x=2](https://tex.z-dn.net/?f=x%3D2)
b) ![x=2](https://tex.z-dn.net/?f=x%3D2)
c) ![x=8](https://tex.z-dn.net/?f=x%3D8)
Part 3)
a) The height of the truss is 12 units
b) The centroid of triangle DEF is 8 units down from D
Step-by-step explanation:
Part 1)
we know that
A <u><em>centroid</em></u> of a triangle is the point where the three medians of the triangle meet. A median of a triangle is a line segment from one vertex to the mid point on the opposite side of the triangle
The centroid divides each median in a ratio of 2:1
Part a) Find the length of the segment AG
we know that
---> the centroid divides each median in a ratio of 2:1
we have
![AD=15\ units](https://tex.z-dn.net/?f=AD%3D15%5C%20units)
substitute
![AG=\frac{2}{3}15=10\ units](https://tex.z-dn.net/?f=AG%3D%5Cfrac%7B2%7D%7B3%7D15%3D10%5C%20units)
Part b) Find the length of the segment GD
we know that
---> the centroid divides each median in a ratio of 2:1
we have
![AD=15\ units](https://tex.z-dn.net/?f=AD%3D15%5C%20units)
substitute
![GD=\frac{1}{3}15=5\ units](https://tex.z-dn.net/?f=GD%3D%5Cfrac%7B1%7D%7B3%7D15%3D5%5C%20units)
Part c) Find the length of the segment CD
we know that
In the right triangle CGD
Applying the Pythagorean Theorem
![CG^2=GD^2+CD^2](https://tex.z-dn.net/?f=CG%5E2%3DGD%5E2%2BCD%5E2)
we have
![CG=13\ units\\GD=5\ units](https://tex.z-dn.net/?f=CG%3D13%5C%20units%5C%5CGD%3D5%5C%20units)
substitute
![13^2=5^2+CD^2](https://tex.z-dn.net/?f=13%5E2%3D5%5E2%2BCD%5E2)
![CD^2=144\\CD=12\ units](https://tex.z-dn.net/?f=CD%5E2%3D144%5C%5CCD%3D12%5C%20units)
Part d) Find the length of the segment GE
we know that
---> the centroid divides each median in a ratio of 2:1
we have
![CG=13\ units](https://tex.z-dn.net/?f=CG%3D13%5C%20units)
substitute
![13=\frac{2}{3}CE](https://tex.z-dn.net/?f=13%3D%5Cfrac%7B2%7D%7B3%7DCE)
![CE=13(3)/2\\CE=19.5\ units](https://tex.z-dn.net/?f=CE%3D13%283%29%2F2%5C%5CCE%3D19.5%5C%20units)
Find the length of the segment GE
![GE=\frac{1}{3}CE](https://tex.z-dn.net/?f=GE%3D%5Cfrac%7B1%7D%7B3%7DCE)
substitute
![GE=\frac{1}{3}19.5\\GE=6.5\ units](https://tex.z-dn.net/?f=GE%3D%5Cfrac%7B1%7D%7B3%7D19.5%5C%5CGE%3D6.5%5C%20units)
Part e) Find the length of the segment GB
we know that
In the right triangle GBD
Applying the Pythagorean Theorem
![GB^2=GD^2+DB^2](https://tex.z-dn.net/?f=GB%5E2%3DGD%5E2%2BDB%5E2)
we have
![GD=5\ units](https://tex.z-dn.net/?f=GD%3D5%5C%20units)
---> D is the midpoint segment CB
substitute
![GB^2=5^2+12^2](https://tex.z-dn.net/?f=GB%5E2%3D5%5E2%2B12%5E2)
![GB^2=169\\GB=13\ units](https://tex.z-dn.net/?f=GB%5E2%3D169%5C%5CGB%3D13%5C%20units)
Part 2) Point L is the centroid of triangle NOM
Find the value of x
Part a) we have
OL=8x and OQ=9x+6
we know that
---> the centroid divides each median in a ratio of 2:1
substitute the given values
![8x=\frac{2}{3}(9x+6)](https://tex.z-dn.net/?f=8x%3D%5Cfrac%7B2%7D%7B3%7D%289x%2B6%29)
solve for x
![24x=18x+12\\24x-18x=12\\6x=12\\x=2](https://tex.z-dn.net/?f=24x%3D18x%2B12%5C%5C24x-18x%3D12%5C%5C6x%3D12%5C%5Cx%3D2)
Part b) we have
NL=x+4 and NP=3x+3
we know that
---> the centroid divides each median in a ratio of 2:1
substitute the given values
![(x+4)=\frac{2}{3}(3x+3)](https://tex.z-dn.net/?f=%28x%2B4%29%3D%5Cfrac%7B2%7D%7B3%7D%283x%2B3%29)
solve for x
![3x+12=6x+6\\6x-3x=12-6\\3x=6\\x=2](https://tex.z-dn.net/?f=3x%2B12%3D6x%2B6%5C%5C6x-3x%3D12-6%5C%5C3x%3D6%5C%5Cx%3D2)
Part c) we have
ML=10x-4 and MR=12x+18
we know that
---> the centroid divides each median in a ratio of 2:1
substitute the given values
![(10x-4)=\frac{2}{3}(12x+18)](https://tex.z-dn.net/?f=%2810x-4%29%3D%5Cfrac%7B2%7D%7B3%7D%2812x%2B18%29)
solve for x
![30x-12=24x+36\\30x-24x=36+12\\6x=48\\x=8](https://tex.z-dn.net/?f=30x-12%3D24x%2B36%5C%5C30x-24x%3D36%2B12%5C%5C6x%3D48%5C%5Cx%3D8)
Part 3)
Part a) Find the altitude of the truss
Let
M ----> the midpoint of segment FE
DM ---> the altitude of the truss
Applying Pythagorean Theorem in the right triangle FDM
![FD^2=FM^2+DM^2](https://tex.z-dn.net/?f=FD%5E2%3DFM%5E2%2BDM%5E2)
substitute the given values
![15^2=9^2+DM^2](https://tex.z-dn.net/?f=15%5E2%3D9%5E2%2BDM%5E2)
![DM^2=225-81\\DM^2=144\\DM=12\ units](https://tex.z-dn.net/?f=DM%5E2%3D225-81%5C%5CDM%5E2%3D144%5C%5CDM%3D12%5C%20units)
therefore
The height of the truss is 12 units
Part b) How far down from D is the centroid of triangle DEF?
we know that
--> the centroid divides each median in a ratio of 2:1
substitute the value of DM
![DG=\frac{2}{3}12=8\ units](https://tex.z-dn.net/?f=DG%3D%5Cfrac%7B2%7D%7B3%7D12%3D8%5C%20units)