Question

Coordinate Geometry: Find the coordinates of the centroid of each triangle with the given vertices. X(5,7), Y(9, -3), Z(13,2). Show Work and Please Help!

Answer 1

Which point of concurrency is the centroid? I think if you define some equations for the corresponding lines that intersect at the centroid, you'll find it.

Answer 2

Ok, Can you help me with that?

Answer 3

Well, the centroid is the intersection of the vertices to the opposite midpoints. We can find the midpoints and then use some Algebra to find the slope and then point-slope form of the equations

Answer 4

Ok so would I use the Slope Formula to find the vertices First?

Answer 5

we already have the vertices... we'd have to use midpoint formula on those given vertices to find our midpoints (there are three midpoints, one for each side XY, YZ, and XZ

\[ \large M_{idpoint}(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}2) \]

Answer 6

X(5,7), Y(9,-3), Z(13,2)
I'll call the midpoints A, B, and C.
M(XY) = A(5+9 / 2, 7+(-3) / 2)
= A(14/2, 4/2)
= A(7,2)

M(YZ) = B(9+13 / 2, -3+2 / 2)
= B(22/2,-1/2)
= B(11,-1/2)

M(XZ) = C(5+13 / 2,7+2 / 2)
= C(18/2,9/2)
= C(9,4 1/2)

Answer 7

Ok is this all of my work for my problem or just for that part?

Answer 8

that was just for the midpoints... we still have to find slope of lines and then find the intersection of two of the lines

Answer 9

Ok, That would be great if you could help! Because I have another I'm stuck on using the Orthocenter

Answer 10

the pairs of vertices that will make the lines from one vertex to the opposite midpoint are AZ, BX, and CY (the vertex that wasn't used to find each midpoint pair together, basically)

If we look at AZ and CY, though, we can see that they are 'special', namely AZ is horizontal and CY is vertical. The equations of those graphs are Y=2 and X=9. These two lines intersect at (X,Y) = (9,2), which would be the centroid.

Answer 11

Ok is that it? (:

Answer 12

yep. :)
I would be doing essentially the same thing for orthocenter, making line equations for each line that intersects the orthocenter (that would be the altitudes of the triangle, which are perpendicular to the sides and go through the opposite vertex) and finding the intersections.

Answer 13

Ok, Can you he;p me with the Orthocenter because I'm not to sure on how to do it either.

Answer 14

are we using the same coordinates as the first problem?

Answer 15

It say's: Coordinate Geometry: Find the coordinates of the orthocenter of each triangle with the given vertices. J(3, -2), K(5, 6), L(9, -2)

Answer 16

its almost like we're doing algebra review in geometry, i think.
the orthocenter is just a fancy name for "intersection of altitudes," which is code for "find the equation of the line perpendiculars of each side of the triangle that intersects the point (vertex), and then you get the system of equations to solve"

Answer 17

Oh ok can you help solve?

Answer 18

first, we'll find the slopes of each side.
assume that the slope equation is in this form for each:
\[ m= \frac{y_2 - y_1}{x_2 - x_1} \]
Slope of JK = y2 - y1 / x2 - x1
= 6 - (-2) / 5 - 3
= 8 / 2
= 4
Slope of KL = y2 - y1 / x2 - x1
= -2 - 6 / 9 - 5
= -8 / 4
= -2
Slope of JL = y2 - y1 / x2 - x1
= -2 - -2 / 9 - 3
= 0 / 6
= 0

Then we can just take their opposite reciprocals for the perpendicular slopes.

Answer 19

Our perpendicular slopes: JK: -1/4, KL: 1/2, and JL: undefined
Each perpendicular will intersect the vertex, so we know at least one point on each line. This opens up our option of point-slope form, using the perpendicular slope and the corresponding vertex.

y - y1 = m(x - x1); Line Perpend. to JK, through L(9,-2)
y + 2 = (-1/4)(x - 9)

y - y1 = m(x - x1); Line Perpend. to KL, through J(3,-2)
y + 2 = (1/2)(x - 3)

Line Perpend. to JL, through K(5,6) => This is a vertical line, so x= something. Of course, if we're going through a point, we'd just use its x-value as the equation.
x = 5.

Since we have a really simple vertical line, let's just use one of our other equations and substitute in the x=5 and solve for y. This will be the intersection point, the Orthocenter!

Answer 20

y + 2 = (-1/4)(5 - 9)
y + 2 = (-1/4)(-4)
y + 2 = 1
y = -1

(5, -1) is the orthocenter

Does this all make sense? I may be making it seem longer than it actually is, but overall its a pretty simple process...

Answer 21

@AccAccessDenied: I so..ooo admire you at thoroughly explanation 0..0
Briefly:
A. Centroid:
1. Find 2 midpoints
2. Find 2 perpendicular slopes
=> Intersection of 2 perpendicular lines through midpoints.
B. Orthorcenter:
Since we already found perpedicular slopes.
=> Intersection point of 2 perpendicular lines through 2 vertices.

Answer 22

mm.. i personally love doing coordinate geometry, i dont even understand why. :P

Answer 23

Ok, I have a couple of more using the centroid formula. If you don't care lol

Answer 24

centroid formula?

Answer 25

Denied, My bad wrong problem lol

Answer 26

mmk... i couldn't exactly tell where each part belonged, so if its another figure, maybe try to write values to the side and draw arrows to where each goes..

Answer 27

Ok, See my teacher assigned me like a big assignment like a (2-40) Even and it's really hard.

Answer 28

maybe your teacher just really wants you to understand it. giving a whole bunch of practice. :P

Answer 29

Yeah I think so, But I know for a fact I need help

Answer 30

so, got your next question you need help on? I'm able to help.

Answer 31

Yeah here it is, Copy and complete each statement for (Triangle Symbol)RST for medians --RM(Suppose to be under), --SL(Suppose to be under), and --TK(Suppose to be under), and centroid J. JT = x(TK).

Answer 32

well, i'd start by drawing the triangle in question, just to get an idea of where everything is:


JT is the distance from the vertex T to the centroid. TK is the entire median.