# Determine m∈R, such as the distance between the points A(2,m) and B(m,−2) is 4.

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It is given that the distance between A(2,m) and B(m,−2) is 4.

The distance between two points (x1, y1) and (x2, y2) is given by sqrt[(x1 - x2)^2 + (y1 - y2)^2]

Substituting the values we have here, we get :

sqrt[(2 - m)^2 + (m + 2)^2] = 4

square both the sides

(2 - m)^2 + (m + 2)^2 = 16

=> 4 - 4m + m^2 + m^2 + 4m + 4 = 16

=> 2m^2 + 8 = 16

=> 2m^2 = 8

=> m^2 = 4

=> m = 2 and m = -2

**The required values of m can be 2 and -2.**

We'll recall the distance formula:

d = sqrt[(xB - xA)^2 + (yB - yA)^2]

Since the coordinates of A and B are given, we'll replace them into the formula:

d = sqrt[(m-2)^2 + (m+2)^2]

We'll expand the binomials:

d = sqrt(m^2 - 4m + 4 + m^2 + 4m + 4)

d = sqrt(2m^2 + 8)

But d = 4 => sqrt(2m^2 + 8) = 4

We'll raise to square both sides:

2m^2 + 8 = 16

2m^2 = 16 - 8

2m^2 = 8

m^2 = 4

m1 = -2 and m2 = 2

The real values of m, such as the distance between the points A(2,m) and B(m,−2) to be 4, are: {-2 ; 2}.