In the Diagram A is the point (-1,1) B is (3,3) and C is (6,2).

For A, I think I know why it is not working properly. You appear to have done everything correctly, so I presume that there is a silly computational error. 

1. Determine the Midpoint of Both Coordinates

The midpoint is the average of both x- and y- coordinates. In algabraic terms, the formula is \(\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\). Let's use it! You used this formula correctly, as I can tell. For completeness, I'll calculate it also.

2. Determine Slope (or Gradient) of the Segment

This also has a formula, luckily, and all we have to do is plug the coordinates in. It is \(m=\frac{y_2-y_1}{x_2-x_1}\), where m is the slope of the segment. Let's just use the formual to determine the slope.

This is not the slope of the perpendicular bisector but rather the slope of the original 2 segments.

3. Determine The Slope of Perpendicular Bisector

A line that is perpendicular to a segment has a slope that is opposite and reciprocal to the slope of the original segment. What does that mean? Well, let's deal with the current slope as its example of \(-\frac{1}{3}\).

I find doing the reciprocal of a number first is generally easier, but the order in which you do the next steps do not matter. In general, the reciprocal of a number follows tthe following rule of \(a\Rightarrow\frac{1}{a}\). Let's apply that now with our slope. \(-\frac{1}{3}\Rightarrow-\frac{1}{\frac{1}{3}}=-3\)

Now, find the opposite. To do this, use the rule that \(a\Rightarrow-a\). \(-3\Rightarrow3\).

Therefore, \(3\) is the slope of the perpendicular bisector. 

If you are interested, here is another method to obtain the slope of the perpendicular bisector. If you take the original slope of the segment and multiply it by the slope of the perpendicular bisector, you will get -1. Let's see this in action!

Look at that! you get the same answer. 

4. Use Point-Slope Form

There are 2 methods again to get to the final equation of the slope, but I generally like point-slope form for some reason, but I will demonstrate both to you, so that you can choose your method. Point-slope form is the following:

\(y-y_1=m(x-x_1)\) where \((x_1,y_1)\) is a point on said segment or line and m is the slope of said segment or line. We know what segment is on this perpendicular bisector; we already calculated it. The perpendicular bisector is a segment that both intersects a segment at its midpoint and at a perpendicular angle. We have already calculated the midpoint of BC, which is \(\left(\frac{9}{2},\frac{5}{2}\right)\). We also know the slope, which is 3. Since we have all that information, let's plug it in!

This is the equation of the perpendicular bisector. I'm not sure where you went wrong, unfortunately, but it seems to me that it is probably some silly computational error. That's all. 

Another method of doing this is using the slope-intercept form from the beginning. Let's see how that would work:

Hopefully, this helped to find the segment that is the perpendicular bisector. 

Now, it is time to do B. Thankfully, this will not be as time-consuming as generating this repsonse has taken me hours. The point you want me to identify is called the circumcenter of the triangle. it is the point located at the center of a circle such that circle contains all the points in a triangle. The circumcenter of a triangle can be found by finding the intersection of 2 of the perpendicular bisectors! Let's do that. We already know 2 lines and their respective equations. They are \(y=3x-11\) and \(y+2x=4\). Let's set up a system of equations.

{ \(y=3x-11\)

{ \(y+2x=4\)

I will use substitution in this case as we already know what y equals.

Now, plug in x into one of the equations. I think that equation 1 will be easier, so I will use that!

Therefore, the circumcenter of the triangle is \((3,-2)\). 

If you would like, click here to view a graph of the triangle, its respective lines that are perpendicular bisectors and the circumcenter.