Question

c) Algebraically, solve the system of equations represented by the medians to determine the location of the centroid of
The coordinates of ∆ are (4,7), (6, −1), and (−2,3).

Answer 1

WRONG AGAIN

thr centroid should be (8/3 , 3)

Answer 2

I even told the bot what the right answer is and it continues with a wrong solution. Hopeless !

Answer 3

To find the centroid, we need to find the point where all three medians intersect. Each median passes through a vertex and the midpoint of the opposite side. We can find the equation of each median using the midpoint formula and point-slope form.

Median 1: Passes through (4,7) and the midpoint of (-2,3) and (6,-1).

Midpoint of (-2,3) and (6,-1): ((-2+6)/2, (3-1)/2) = (2,1)

Midpoint formula: ((4+2)/2, (7+1)/2) = (3,4)

Slope: (-1-3)/(6-(-2)) = -4/8 = -1/2

Equation of median 1: y - 7 = (-1/2)(x - 4)

y - 7 = (-1/2)x + 2

y = (-1/2)x + 9

Median 2: Passes through (6,-1) and the midpoint of (4,7) and (-2,3).

Midpoint of (4,7) and (-2,3): ((4-2)/2, (7+3)/2) = (1,5)

Midpoint formula: ((6+1)/2, (-1+5)/2) = (3,2)

Slope: (2+1)/(-1-6) = 3/-7

Equation of median 2: y + 1 = (3/-7)(x - 6)

y + 1 = (-3/7)x + 18/7

y = (-3/7)x + 11/7

Median 3: Passes through (-2,3) and the midpoint of (4,7) and (6,-1).

Midpoint of (4,7) and (6,-1): ((4+6)/2, (7-1)/2) = (5,3)

Midpoint formula: ((-2+5)/2, (3+3)/2) = (3,3)

Slope: (3-3)/(6-(-2)) = 0

Equation of median 3: x = -2

Now we have three equations:

y = (-1/2)x + 9

y = (-3/7)x + 11/7

x = -2

We can solve for x and y by substituting x = -2 into the first two equations:

y = (-1/2)(-2) + 9 = 10

y = (-3/7)(-2) + 11/7 = 23/7

Therefore, the centroid of triangle ∆ is at the point (-2,10/3), or approximately (-2,3.33).

Answer 4

I apologize for my mistake. Here's the corrected solution:

To find the centroid, we first need to find the midpoint of each side of the triangle.

Midpoint of (4,7) and (6,-1): ((4+6)/2, (7-1)/2) = (5,3)

Midpoint of (6,-1) and (-2,3): ((6-2)/2, (-1+3)/2) = (2,1)

Midpoint of (-2,3) and (4,7): ((-2+4)/2, (3+7)/2) = (1,5)

Now, we can use the midpoint formula to find the equation of the line passing through each midpoint and the opposite vertex.

Line passing through (5,3) and (-2,3) (opposite vertex: (4,7)):

Slope = (7-3)/(4-5) = -4

y - 3 = -4(x - 5)

y = -4x + 23

Line passing through (2,1) and (4,7) (opposite vertex: (-2,3)):

Slope = (3-1)/(-2-4) = -1/3

y - 1 = -(1/3)(x - 2)

y = -(1/3)x + 7/3

Line passing through (1,5) and (6,-1) (opposite vertex: (-2,3)):

Slope = (3 - 5)/(6 - 1) = -2/5

y - 5 = (-2/5)(x - 1)

y = (-2/5)x + 11/5

To find the intersection point of the three lines, we solve for the x-coordinate and then substitute into any of the equations to find the y-coordinate.

-4x + 23 = -(1/3)x + 7/3

-11x/3 = -16/3

x = 16/11

y = -4(16/11) + 23 = 3

Therefore, the centroid of the triangle is at the point (16/11, 3), which is approximately (1.45, 3) in decimal form.

Answer 5

I apologize for the inconvenience and frustration this has caused you. I am here to assist and provide accurate solutions to the best of my ability. I will double-check my work and make sure to provide the correct solution moving forward to avoid any further issues.