Answer:
[tex]\tt m = 6[/tex] or [tex]\tt m = -2[/tex]
Step-by-step explanation:
The distance formula between two points [tex]\tt (x_1, y_1)[/tex] and [tex]\tt (x_2, y_2)[/tex] in a coordinate plane is given by:
[tex]\tt d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/tex]
In this case, we have points A(2, -1) and B(m, -5), and the distance is given as [tex]\tt 4\sqrt{2}[/tex].
Substituting the coordinates into the distance formula, we get:
[tex]\tt 4\sqrt{2} = \sqrt{(m - 2)^2 + (-5 - (-1))^2}[/tex]
Simplify the equation:
[tex]\tt 4\sqrt{2} = \sqrt{(m - 2)^2 + (-4)^2}[/tex]
Square both sides to eliminate the square root:
[tex]\tt (4\sqrt{2} )^2= (\sqrt{(m - 2)^2 + (-4)^2})^2[/tex]
[tex]\tt 32 = (m - 2)^2 + 16[/tex]
Subtract 16 from both sides:
[tex]\tt 32 -16= (m - 2)^2 + 16-16[/tex]
[tex]\tt 16 = (m - 2)^2[/tex]
Now, take the square root of both sides, considering both the positive and negative roots:
[tex]\tt 4 = m - 2 \quad \text{or} \quad -4 = m - 2[/tex]
For the positive root:
[tex]\tt m - 2 = 4[/tex]
Add 2 to both sides:
[tex]\tt m - 2 +2 = 4+2[/tex]
[tex]\tt m = 6[/tex]
For the negative root:
[tex]\tt m - 2 = -4[/tex]
Add 2 to both sides:
[tex]\tt m - 2 +2= -4+2[/tex]
[tex]\tt m = -2[/tex]
So, there are two possible values for [tex]\tt m[/tex]:
[tex]\tt m = 6[/tex] or [tex]\tt m = -2[/tex].
