How to Find the Third Vertex of an Equilateral Triangle Given the Coordinates of Its Two Vertices

The musical triangle is in the shape of an equlateral triangle

An equilateral triangle is a three-sided figure with all the sides equal. When all the sides are equal, all the angles must also be equal, and, because the total of the three angles is 180 degrees, each angle must be 60 degrees. You can use these facts to draw an equilateral triangle and get the third vertex. There are actually two choices for that third vertex.

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Graph paper
Protractor

Instructions

- 1
  
  Draw a horizontal and a vertical axis. These should be near the middle of your graph paper.
- 2
  
  Make check marks. They should cover the range of the vertices you are given. For example, if the two vertices are (4, 10) and (6, 12) make sure the x-axis (the horizontal one) covers the range from 4 to 6 and the y-axis (the vertical one) covers 10 to 12.
- 3
  
  Mark the two vertices.
- 4
  
  Draw a straight line between the two vertices.
- 5
  
  Mark an angle of 60 degrees (in either direction) from one end of the line.
- 6
  
  Measure the first line and draw another line of the same length at the marked angle.
- 7
  
  Draw a vertical line from the new point to the x-axis. The point where the line crosses the axis is the x coordinate of the third vertex.
- 8
  
  Draw a horizontal line from the new point to the y-axis. The point where the line crosses the y-axis is the y-coordinate of the third vertex.