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

Question

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

Answer 1

the bot is really wrong!!!

let A be the midpoint of LM, A is (5,3)
notice the median NA is horizontal, so y = 3

let the midpoint of NM be B, B is (2,1), and the slope of LB = 3
for median LB: y-1 = 3(x-2) or y = 3x-5
sub in y=3
3x-5 = 3
3x=8
x = 8/3, and we know y = 3

the centroid is (8/3 , 3)

Verify by using short-cut to find centroid
for the x, add the x's , then divide by 3
(-2 + 4 + 6)/3 = 8,3
for the y, add the y's, then divide by 3
(3+7-1)/3 = 3
so centroid is (8/3 , 3)

Answer 2

I apologize for the incorrect response earlier. Thank you for correcting the solution and providing the proper method. Your explanation is correct, and the centroid of triangle LMN is indeed at (8/3, 3).

Answer 3

To find the centroid of a triangle, you can use the formula that states the centroid is the average of the three vertices. In other words, the coordinates of the centroid are the averages of the coordinates of the three vertices.

Let's start by finding the coordinates of the centroid, which we will label as G(x, y). We will use the coordinates of points L(4, 7), M(6, -1), and N(-2, 3).

To find the x-coordinate of the centroid (xG), we can take the average of the x-coordinates of points L, M, and N:

xG = (xL + xM + xN) / 3

Substituting the coordinates:

xG = (4 + 6 + (-2)) / 3 = 8 / 3

So, the x-coordinate of the centroid is 8/3.

Similarly, to find the y-coordinate of the centroid (yG), we can take the average of the y-coordinates of points L, M, and N:

yG = (yL + yM + yN) / 3

Substituting the coordinates:

yG = (7 + (-1) + 3) / 3 = 9 / 3

So, the y-coordinate of the centroid is 3.

Therefore, the coordinates of the centroid G are (8/3, 3).

Answer 4

We first find the equations of the medians by using the midpoint formula.

The median from L to the midpoint of MN:

Midpoint of MN = (1/2)(M+N) = (1/2)((6, -1) + (-2, 3)) = (2, 1)

Midpoint Formula: [(6+(-2))/2, (-1+3)/2] = [2,1]

Equation of the median from L:

slope = (1-7) / (2-4) = -3

Using point-slope form: y - 7 = -3(x-4)

Simplifying: y = -3x + 19

The median from M to the midpoint of LN:

Midpoint of LN = (1/2)(L+N) = (1/2)((4,7) + (-2,3)) = (1,5)

Midpoint Formula: [(4+(-2))/2, (7+3)/2] = [1,5]

Equation of the median from M:

slope = (5-(-1)) / (1-6) = -4/5

Using point-slope form: y - (-1) = (-4/5)(x-6)

Simplifying: y = (-4/5)x + (14/5)

The median from N to the midpoint of LM:

Midpoint of LM = (1/2)(L+M) = (1/2)((4,7) + (6, -1)) = (5,-2)

Midpoint Formula: [(4+6)/2, (7-1)/2] = [5,-2]

Equation of the median from N:

slope = (-2-3) / (5-(-2)) = -1/7

Using point-slope form: y - 3 = (-1/7)(x+2)

Simplifying: y = (-1/7)x + (23/7)

Now we need to find the point where all three medians intersect - this is the centroid. We can solve the system of equations:

y = -3x + 19

y = (-4/5)x + (14/5)

y = (-1/7)x + (23/7)

To solve for x, we set the second and third equations equal:

(-4/5)x + (14/5) = (-1/7)x + (23/7)

Multiplying both sides by 35 to eliminate the denominators:

-28x + 98 = -5x + 115

Adding 28x to both sides and subtracting 98 from both sides:

93 = 23x

x = 93/23

To find y, we can use any of the three equations - let's use the first one:

y = -3(93/23) + 19

y = (2/23)

Therefore, the centroid of triangle LMN is (93/23, 2/23).