The position vectors A and B with respect to the origin O,(-8,5) are (12,-5)and respectively. Point M is the midpoint of AB and N is the midpoint of OA.

(a) Find:
(i) The coordinates of N and M;
(ii) The magnitude of NM,
(b) Express vector NM in term OB.
(c) Point P maps onto P' by a translation(-5,8).Give that OP = OM + 2MN, find the coordinates of P'.

I will be happy to check your thinking on this,it seems straightforward to me.

A)

M= AB
= (-8+ 12)/2, (5 - (-5))/2
= (2, 0)
N = OA
(0 + -8)/2, (0 + 5)/2
= (-4, 2.5)
|NM| = 36 + 6.25
= 42.25
NM = 6.5

B) NM = OB
= (12 + 0)/2, (-5 + 0)/2
= (6, -2.5)
|NM| = 10(10) + (-5)(-5)
= 100 + 25
= 125
= 11.18
= 11.18/6.5
= 1.82

C)

A=(-8,5)

B=(12,-5)

M = (A+B)/2 = (2,0)
N = A/2 = (-4,5/2)
NM∥OB and NM=OB/2

OP=OM+2MN = OM-OB
OP' = OP+(-5,8)

I am still not get a solution C

(1, 0) - (6, -2.5)

= (-5, . 2.5)

(a)

(i) The coordinates of N can be found by finding the midpoint of OA:

N = (1/2)(O + A)
= (1/2)(-8,5 + 12,-5)
= (1/2)(4,0)
= (2,0)

The coordinates of M can be found by finding the midpoint of AB:

M = (1/2)(A + B)
= (1/2)(12,-5 + -8,5)
= (1/2)(4,0)
= (2,0)

Therefore, N = (2,0) and M = (2,0).

(ii) The magnitude of NM can be found using the distance formula:

NM = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((2 - 2)^2 + (0 - 0)^2)
= sqrt(0 + 0)
= sqrt(0)
= 0

Therefore, the magnitude of NM is 0.

(b) We can express vector NM in terms of vector OB as follows:

NM = MO + OB
= -OM + OB
= -2MN + OB

Therefore, vector NM can be expressed as -2MN + OB.

(c) Given that OP = OM + 2MN, we can solve for P':

P' = OP - OM - 2MN
= (-5,8) - (2,0) - 2(2,0)
= (-5,8) - (2,0) - (4,0)
= (-5,8) - (6,0)
= (-11,8)

Therefore, the coordinates of P' are (-11,8).

To find the coordinates and vectors in this problem, we'll make use of basic vector operations and formulas.

(a) (i) To find the coordinates of N and M, we need to calculate the midpoint of the given vectors.
The midpoint formula states that the coordinates of the midpoint of two vectors (x1, y1) and (x2, y2) are ((x1+x2)/2, (y1+y2)/2).

Given vectors A=(-8,5) and B=(12,-5):
Coordinates of N = ((-8+0)/2, (5+0)/2) = (-4/2, 5/2) = (-2, 2.5)
Coordinates of M = ((12-8)/2, (-5+5)/2) = (4/2, 0/2) = (2, 0)

(ii) To find the magnitude of vector NM, we can use the distance formula between two points.
The distance formula states that the magnitude of a vector with coordinates (x1, y1) and (x2, y2) is sqrt((x2-x1)^2 + (y2-y1)^2).

Coordinates of N = (-2, 2.5)
Coordinates of M = (2, 0)

Magnitude of NM = sqrt((2-(-2))^2 + (0-2.5)^2) = sqrt((4)^2 + (-2.5)^2) = sqrt(16 + 6.25) = sqrt(22.25) = 4.71 (approximately)

(b) To express vector NM in terms of vector OB, we need to find the components of vector NM and vector OB.

Coordinates of N = (-2, 2.5)
Coordinates of M = (2, 0)
Coordinates of O = (0, 0)
Coordinates of B = (12, -5)

Components of vector NM = (x2-x1, y2-y1) = (2-(-2), 0-2.5) = (4, -2.5)

Components of vector OB = (x2-x1, y2-y1) = (12-0, -5-0) = (12, -5)

Therefore, vector NM in terms of vector OB is NM = OB - OM = (12, -5) - (4, -2.5) = (8, -2.5).

(c) Given that OP = OM + 2MN, we can find the coordinates of P' by using the translation vector (-5, 8).

Coordinates of O = (0, 0)
Coordinates of M = (2, 0)
Coordinates of N = (-2, 2.5)

Using OP = OM + 2MN, we can substitute the coordinates into the equation:

(xp, yp) = (xm, ym) + 2(xn, yn)
(xp, yp) = (2, 0) + 2*(-2, 2.5)
(xp, yp) = (2, 0) + (-4, 5)
(xp, yp) = (-2, 5)

Finally, applying the translation vector (-5, 8), we get:
Coordinates of P' = (-2-5, 5+8) = (-7, 13)

Therefore, the coordinates of P' are (-7, 13).