Determining the 3rd vertex of an Equilateral and Right angled isoceles triangle.

Question

I am really having problems solving the following problems:

If (x1,y1) and (x2,y2) are the coordinates of the two vertices on the hypotenuse of a right angled isosceles triangle then the coordinates of the 3rd vertex are?
Similarly-If (x1,y1) and (x2,y2) are the coordinates of the two vertices of an Equilateral Triangle then the coordinates of the 3rd vertex are?
The answer to 1 and 2 respectively are-

(x1+x2±(y1−y2)2,y1+y2±(x1−x2)2)
And

(x1+x2±3√(y1−y2)2,y1+y2±3√(x1−x2)2)
How do I solve this problem?

I tried attempting it by using the fact that the triangle is isosceles and right angled in the first case. So I applied pythagoras theorem but failed to get this answer it was a complicated equation. Secondly I tried multiplying the slopes of the legs of the triangle to get -1 but even this attempt failed.

It was similar with the equilateral triangle problem.

I guess if I understand how to do even one easily the second should bit be difficult.

Please help.

Answer 1

This is an interesting problem.  I will do my best to help explain this one to you.

 

 

Part 1

 

 

A right angled isosceles triangle has two legs of equal length and the angle formed by those legs is 90 degrees.  When we have a right triangle, can use the Pythagorean theorem. 

 

If the distance of one leg is  (x3 - x1) and the distance of another leg is (y2 - y3), then the hypotenuse is

 

hypotenuse = √((x3 - x1)² + (y2 - y3)²)

 

I came up with his formula by drawing a triangle and labeling them using the coordinates in terms of x and y.

 

Notice that x3 and y3 are in both of the legs. This is because the legs share a common point which is the third vertex. Also, we use an x-coordinate of one point of the hypotenuse and a y-coordinate on the other point of the hypotenuse to find the length of the legs.

 

Since the legs are equal in length, we can say that

 

x3 - x1 = y2 - y3

 

Now, lets solve for x3 and y3 using this equation.

 

x3 = y2 - y3 + x1

 

x3 - x1 - y2 = -y3

-x3 + x1 + y2 = y3

 

If (x3, y3) is the vertex, then the coordinate of the vertex is

 

(y2 - y3 + x1 ,  -x3 + x1 + y2)

 

If you know this coordinate, then you can find the others by reflecting over the x and y-axis.

 

Does this make logical sense?

 

 

Part 2

 

An equilateral triangle has sides of equal length all around.  If we draw a line from one vertex of the triangle to the side opposite of that vertex, with the line being perpendicular to that side, then we will have to right triangles.  As a result, one of the legs will be half of its original length.

 

Combine the information here with the method use from part 1 to find the coordinates.