Construct a Triangle from Three Sides

Constructing a Triangle from Three Given Line Segments Using a Compass and Straightedge

To construct a triangle when given three line segments (let’s call them A, B, and C), follow these steps:

  1. Check the Triangle Inequality:
    • Ensure that the sum of any two segments is greater than the third segment. That is:
      • A+B>CA + B > CA+B>C
      • A+C>BA + C > BA+C>B
      • B+C>AB + C > AB+C>A
    • If this condition is not met, a triangle cannot be formed.
  2. Draw the First Segment:
    • Use a straightedge to draw one of the three segments, say A, as a base. Label its endpoints as P and Q.
  3. Construct an Arc for the Second Segment:
    • Place the compass at P and set its radius to the length of B.
    • Draw an arc that represents all possible locations for the second vertex of the triangle.
  4. Construct an Arc for the Third Segment:
    • Place the compass at Q and set its radius to the length of C.
    • Draw another arc that represents all possible locations for the same vertex.
  5. Locate the Third Vertex:
    • The intersection of the two arcs determines the third vertex of the triangle. Label this point as R.
  6. Complete the Triangle:
    • Use the straightedge to connect P to R and Q to R.

This method ensures an exact construction of the triangle, provided the given segment lengths satisfy the triangle inequality theorem.

Animation: