Mathematics, 22.06.2019 13:00 SoUpsup

The three lines 8x+3y=12, 6y-7x=24, x+9y+33=0 intersect to form a triangle. find the coordinates of its centroid

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Answer from: mjwaple57

coordinates of its centroid=(-1,-1)

Step-by-step explanation:

The given three lines are:

8x+3y=12, (1)

6y-7x=24 (2)

and x+9y+33. (3)

Multiplying equation (1) by 2 and subtract the equation (2), we get

23x=0⇒x=0,

Putting the value of x=0 in equation (2), we get

6y=24

y=4

Now, multiply the first equation by 3 and subtract the equation (3), we get

23x=69⇒x=3

Putting the value of x=3 in equation (1), we get

24+3y=12

3y=-12

y=-4

Also, multiplying the equation (3) by 7 and add the equation (2), we get

69y=-207

y=-3

Putting the value of y=-3 in equation (2), we get

6(-3)-7x=24

-18-24=7x

x=-6

So the centroid is located at (the average of the three vertices):

$x=\frac{0+3-6}{3}=\frac{-3}{3}=-1$ and $y=\frac{4-4-3}{3}=\frac{-3}{3}=-1$ .

Thus, the coordinates of its centroid=(-1,-1)

Answer from: dolphinkina35

(a) Please see the first figure attached

(b) The coordinates of the centroid are $G(\frac{2}{3}, \frac{m+n}{3} )$

(c) due to the definition of the centroid of a triangle, this will always lie inside the triangle, therefore, for any value of and , the centroid will lie

Step-by-step explanation:

Hi, let us first solve part (a). Since for any given values of and we will obtain two linear functions:

y=-mx and

y=-nx

with $0\leq x\leq 1$ we can assure that our region is going to be a triangle. To see this, please take a look at the plot I generated using Wolfram. In this case, I have used two specific values for m and n but keeping the condition $0\leq n\leq m$ .

Now, for part (b) let me start remembering what the centroid is: the centroid of a triangle is the point where the three medians of the triangle meet. And a median of a triangle is a line segment from one vertex to the midpoint on the opposite side of the triangle (see the second figure where the medians are depicted in red and the centroid of the triangle, G is depicted in blue). For a given triangle $\bigtriangleup \rm{ABC}$ , the coordinates of its centroid are given by:

$G_x=\frac{A_x+B_x+C_x}{3}$ and $G_y=\frac{A_y+B_y+C_y}{3}$

Now let's apply this to our problem. Take a look at the first figure. The vertex A has clearly coordinates (0,0) for any value of and since the two lines have their intersection with y-axis in this point.

To obtain the coordinates of and , let's use the given functions and the fact that the coordinate x is limited to 1. Then, we have:

For A:

y=-mx then, when x=1 , substituting in the formula y=-m

For B and doing the same as for A:

y=-nx then, when x=1 , substituting in the formula y=-n

Thus, the coordinates of the vertices of the triangle are: A(1, m) , B(1,3) and C(0,0) and the coordinates of the centroid are:

$G_x=\frac{A_x+B_x+C_x}{3} = G_x=\frac{1+1+0}{3}\\G_x=\frac{2}{3}$

and

$G_y=\frac{A_y+B_y+C_y}{3}=\frac{m+n+0}{3}\\G_y=\frac{m+n}{3}$ .

Summarizing: the coordinates of the centroid of the region are $G(\frac{2}{3}, \frac{m+n}{3} )$

Now, for part (c), due to the definition of the centroid of a triangle, this will always lie inside the triangle, therefore, for any value of and , the centroid will lie. Other important points of the triangle, like the orthocentre and circumcentre, can lie outside in obtuse triangles. In right triangles, the orthocentre always lies at the right-angled vertex.

Let 5 be the region that lies between the curves y=− xm ; y= − xn ; 0 < x < 1 where m and

Answer from: goreimani9

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Answer from: maddietomlinson113

The centroid is the average of the three vertices’ coordinates, so it’s ((-1+1+5)/3, (3–1+1)/3) or (5/3, 1).

Answer from: jahazielBrainly15006

The coordinates of centroid are: (10/3, 3)

Step-by-step explanation:

The formula for calculating centroid of a triangle is:

$(x,y) = (\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3})$

Here (x1,y1) are the coordinates of first vertex

(x2,y2) are the coordinates of second vertex

(x3,y3) are the coordinates of third vertex

Given:

G(-2,5) = (x1,y1)

H(6,5) =(x2,y2)

J(6,-1) = (x3,y3)

Let I be the centroid of the triangle

Putting the values in the formula

$I(x,y) = (\frac{-2+6+6}{3},\frac{5+5+-1}{3})\\= (\frac{10}{3},\frac{9}{3})\\=(\frac{10}{3},3)$

The coordinates of centroid are: (10/3, 3)

Keywords: Centroid, Triangle

Learn more about centroid at:

#LearnwithBrainly

Answer from: itskarmaboo

Given the coordinates of the three vertices of a triangle ABC,
the centroid O coordinates are given by :

Ox=Ax+Bx+Cx/3
= 5+9+13/3
= 9
Oy=Ay+By+Cy/3
= 7-3+2/3
= 2

Answer from: naomicervero

Given the line 12x + 9y = 36

the line intersects the x-axis when y = 0,

i.e. 12x = 36 ⇒ x = 3

Thus the x-intecept is (3, 0)

the line intersects the y-axis when x = 0,

i.e. 9y = 36 ⇒ y = 4

Thus, the y-intercept is (0, 4)

Thus, the coordinates of the vertices of the triangle are (0, 0), (0, 4) and (3, 0)

The midpoint of the side of the rectangle joining points (0, 0) and (0, 4) is (0, 2) and the equation of the median joining points (0, 2) and (3, 0) is given by

$\frac{y-2}{x} = \frac{-2}{3} \\ \\ \Rightarrow3(y-2)=-2x \\ \\ \Rightarrow y=- \frac{2}{3} x+2$

The midpoint of the side of the rectangle joining points (0, 0) and (3, 0) is (1.5, 0) and the equation of the median joining points (1.5, 0) and (0, 4) is given by

$\frac{y}{x-1.5} = \frac{4}{-1.5} \\ \\ \Rightarrow1.5y=-4(x-1.5)=-4x+6 \\ \\ \Rightarrow y=- \frac{8}{3} x+4$

The point of intersetion of the two medians is given by

$- \frac{2}{3} x+2=- \frac{8}{3} x+4 \\ \\ \Rightarrow2x=2 \\ \\ \Rightarrow x=1 \\ \\ \Rightarrow y=-\frac{2}{3}+2=\frac{4}{3}$

Therefore, the centroid of the given triangle is located at point $\left(1, \frac{4}{3} \right)$

Answer from: mattydoug4818

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Answer from: maria5633

The graph of the region is attached. The coordinates of the centroid are (5/3, 1).

The coordinates of the centroid are found by averaging the coordinates of the region;

Oₓ = (Aₓ+Bₓ+Cₓ)/3 = (0+1+4)/3 = 5/3
O(y) = (A(y) + B(y) + C(y)) = (0+3+0)/3=3/3=1

Sketch the region bounded by the curves y=3x3, x+y=4, and y=0. find the coordinates of the centroid.