Triangle inequality – joining three segments
MathematicsPlane GeometryAges 12–13
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Plane Geometry
Three segments a, b, c form a triangle if and only if each side is shorter than the sum of the other two. Drag the sliders or drag vertices A and C to change the lengths: when one segment is too long, the other two laid end to end still cannot meet, and the simulation shows the gap clearly. When a triangle can be formed, the angles appear so you can see the largest side facing the largest angle.
Lesson: Relationship between angles and opposite sides; the triangle inequality
What it shows
Three segments can form a triangle only if each one is shorter than the sum of the other two: a < b + c, b < a + c and c < a + b. In practice it is enough to check the longest side against the sum of the other two. When the longest side equals that sum, the segments lie flat and the three points are collinear; when it is longer, a gap remains. Once a triangle exists, the angles are computed with the law of cosines and show that the largest angle always lies opposite the longest side, and the smallest angle opposite the shortest.
How to use
Set the three sides with the sliders or by dragging vertices A and C, and have students predict before each change whether a triangle will form. Press Example that fails to load 9, 4 and 3 cm, then shorten side a step by step until the gap closes and a triangle appears. Keep Show angle measures on and increase one side at a time to see which angle grows.
Parameters you can change
- Side a = BC 1–10 cm
- Side b = CA 1–10 cm
- Side c = AB 1–10 cm
- Show angle measures
Questions to explore
- Why is it enough to compare only the longest side with the sum of the other two?
- With b = 4 cm and c = 3 cm, which values of a allow a triangle to form?
- As you lengthen side a, which angle of the triangle gets larger?
