Question

How exactly am i supposed to locate? please help​

Answer 1

We can use the vector addition property to find the coordinates of the points M, N, and I.

Let

`\bold{ A = (x_1, y_1), B = (x_2, y_2), C = (x_3, y_3), and\: D = ((x_2+x_3)/(2), (y_2+y_3)/(2))}`

To find M, we use the fact that MA + MB = 0. Therefore, we have:

MA = -MB

A - M = -(B - M)

A - M = -B + M

2M = A + B

M =
`(A + B)/(2)`

So, the coordinates of point M are

`\bold{((x_1+x_2)/(2), (y_1+y_2)/(2))}`

To find N, we use the fact that 3NA + NC = 0. Therefore, we have:

3NA = -NC

3A - N = -(C - N)

3A - N = -C + N

4N = 3A + C

N =
`( 3A + C)/(4)`

So, the coordinates of point N are
`\bold{((3x_1+x_3)/(4),( 3y_1+y_3)/(4))}`

To find I, we use the fact that IM + 2IN = 0. Therefore, we have:

IM = -2IN

M - I = -2(N - I)

M - I = -2N + 2I

3I = M + 2N

I =
`((M + 2N))/(3)`

Substituting the values of M and N that we found earlier, we get:

I = (A + B + 2(3A + C)/4)/3

Simplifying this expression, we get:

I =
`((7A + 2B + 2C))/(12)`

So, the coordinates of a point I are
`\bold{((7x_1+2x_2+2x_3)/(12), (7y_1+2y_2+2y_3)/(12)).}`