In an XY plane, How can you find the the other vertice/s of an equilateral triangle when 2 vertices are given?

Question

For example:
P1 (-2,4)
P2 (9,-6)
P3 (x,y), note that there are 2 P3's bcoz they can be found on both sides of the given segment

yes, I definitely know the distance. but the slopes of the other 2 segment is what I don't know.

c'mon, our teacher doesn't even know the answer...

Answer 1

One way to solve it is:

You know the distance the point must have from each of P1 and P2. That tells you two quadratic equations in two variables. Simply subtracting one from the other gives you a linear relationship between x and y. Substituting that back into one of the quadratics gives you a quadratic equation in one variable.

And you surely know how to solve that!

Answer 2

By symmetry, the possibilities will lie on the perpendicular bisector of line P1P2. Additionally, if you construct a circle centered at either P1 or P2 with radius equal to the distance between P1 and P2, you can find the two values of P3 by solving for the intersection points of the perpendicular bisector and the circle.

Or you could find the intersection points of two circles centered at P1 and P2 each with radius equal to the distance between P1 and P2. These should give you the possibilities for the third vertex.

Answer 3

The midpoint betweeen P1 and P2 is (3.5,-1). The slope of the perpendicular line is 10/11.

Answer 4

Equilateral means all sides the same. So if you know the distance between two of the points, then you should be able to figure out the distance between the other point.

Answer 5

Write down your reasoning and work and I'll tell you if its right. Hint: Draw a picture and label everything you know from the question.

Answer 6

Pay attention in class and you'll know how to do it.