Problem

Consider a triangle with vertices (1, 1), (9, 7), and (3, 11). Do the perpendicular bisectors of the sides of the triangle intersect in a single point?

Solution

To get a handle on this problem, we first draw a diagram to organize the information we have and the information we will need to find in order to solve this problem.

For convenience, we call the sides of the triangle S₁ (blue in the diagram), S₂ (cyan), and S₃ (magenta). We will first need to find the equation of the perpendicular bisector of each side. The information we need for that is the midpoint and slope of each side.

After we have the equations of the perpendicular bisectors, we need to check if they intersect. One way to do this is to take two of the lines and find their point of intersection, then check if this point lies on the third line. If it does, then we have successfully shown that all three lines intersect in a single point.

Let's find the perpendicular bisectors of the sides, one at a time.

S₁:

Slope:

Midpoint:

Therefore the perpendicular bisector to S₁ has slope –1/5 and passes through the point (2, 6).

The equation of the line is

and we solve for c by plugging in the point (2, 6):

So the equation of the perpendicular bisector is

S₂:

Slope:

Midpoint:

Slope of perpendicular bisector:

Equation of perpendicular bisector:

Solve for c using midpoint:

Final equation:

S₃:

Slope:

Midpoint:

Slope of perpendicular bisector:

Equation of perpendicular bisector:

Solve for c using midpoint:

Final equation:

As mentioned above, the next step is to find the point of intersection of two of the lines. We'll choose the bisectors of S1 and S2, setting the right hand sides equal to each other, and solving for x:

Multiply by 15 to clear the denominators.

Solve for x.

Substitute this value into either of the two equations to find y.

So the intersection point of the first two bisectors is

Does this point lie on the third bisector? Plug these in for x and y.

Yes. Therefore, all three perpendicular bisectors of the triangle do indeed intersect in a single point, (64/17, 96/17).