In the diagram below, the point $\left(x, y\right)$ is the same distance from $\left(−3, 4\right)$ as it is from $\left(2, −1\right)$. There are many possibilities for $\left(x, y\right)$. Algebraically, what do they have in common?

1. What is the distance from $\left(x, y\right)$ to $\left(−3, 4\right)$?

   Hint (a):

   Use the distance formula.

   Answer (a):

   $\sqrt{(x + 3)^2+(y - 4)^2}$

2. What is the distance from $\left(x, y\right)$ to $\left(2, −1\right)$?

   Hint (b):

   See part (a).

   Answer (b):

   $\sqrt{(x - 2)^2+(y + 1)^2}$

3. Write the equation that states that the two expressions in (a) and (b) are equal and simplify.

   Step 1(c):

   $\sqrt{(x + 3)^2+(y - 4)^2} = \sqrt{(x - 2)^2+(y + 1)^2}$

   Step 2(c):

   $\left(x + 3\right)^{2} + \left(y − 4\right)^{2} = \left(x − 2\right)^{2} + \left(y + 1\right)^{2}$

   Step 3(c):

   $x^{2} + 6x + 9 + y^{2} − 8y + 16 = x^{2} − 4x + 4 + y^{2} + 2y + 1$

   Step 4(c):

   $10y = 10x + 20$

   Answer (c):

   $y = x + 2$

4. What is the specific name of the geometric object represented by your equation from part (c)?

   Answer (d):

   Perpendicular bisector of the segment.