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Triangle Inequality theorem via construction

20 practice questions 2 video lessons Theory + worked examples

The Triangle Inequality by Construction

California Geometry • Standard G-CO.12 • Geometric Constructions

The Triangle Inequality by Construction is a topic in Geometric Constructions in the California Common Core State Standards. It is aligned to Standard G-CO.12, which requires students to use formal constructions to explore when three lengths form a triangle.

Constructing a triangle from three lengths succeeds only when each pair of sides sums to more than the third — a hands-on view of the triangle inequality.

California Geometry › Geometric Constructions › The Triangle Inequality by Construction  —  Standard G-CO.12

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Practice questions

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Watch 2 video(s)
  • Triangle inequality using circle | A Tale of Three Intersecting Lines| NCERT G7| Math | Khan Academy Watch
  • Constructing Triangles Using Side Lengths (Triangle Inequality Theorem) Watch
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Theory

To build a triangle from three lengths, draw the longest as a base, then swing an arc of each other length from an endpoint. The third vertex is where the arcs meet.

The arcs meet only when the two shorter sides can reach across the base — that is, when their sum exceeds the base. This is a visual proof of the Triangle Inequality Theorem.

Arcs meet \(\Leftrightarrow\) a triangle exists \(\Leftrightarrow\) \(a+b>c\) for the longest side \(c\).
Constructing a triangle (arcs meet) When the two arcs from the endpoints of the base reach each other, the third vertex exists and a triangle can be built. A B arcs meet: triangle exists
The arcs meet: the third vertex exists, so a triangle can be built.
No triangle (arcs fall short) If the two shorter sides cannot reach each other over the longest side, no triangle exists. arcs fall short: no triangle gap
The arcs fall short: no triangle is possible.

The existence condition:

\[\text{triangle exists}\iff a+b>c\ \ (\text{for the longest side } c)\]
a triangle exists exactly when the two shorter sides sum to more than the longest
Equality \(a+b=c\) gives a flat, degenerate “triangle” — the arcs just touch on the base.

How to construct (or rule out) a triangle

  1. Draw the longest side as the base.
  2. Swing an arc of the second length from one endpoint.
  3. Swing an arc of the third length from the other endpoint.
  4. If they meet, connect the vertex; if not, no triangle exists.
Example 1 — When do the arcs meet?
In the construction, when do the two arcs from the base endpoints intersect?
Solution

When the two shorter sides together are longer than the base — the triangle inequality.

\(a+b\)\(>\)\(c\)
when the sum of the two shorter sides exceeds the base
Example 2 — A valid set
Can a triangle be constructed with sides \(6,7,8\)?
Solution

Check the two smallest against the largest.

\(6+7\)\(=\)\(13>8\ \checkmark\)

Yes — the arcs will meet.

yes, 6, 7, 8 can be constructed
Example 3 — A failing set
Can a triangle be constructed with sides \(3,4,9\)?
Solution

Test the two shorter sides.

\(3+4\)\(=\)\(7<9\)

No — the arcs fall short, so no triangle exists.

no, 3, 4, 9 cannot be constructed
Example 4 — What the construction shows
What does the compass construction demonstrate about three lengths?
Solution

It gives a visual proof of the Triangle Inequality Theorem: a triangle exists exactly when each side is shorter than the sum of the other two.

the construction demonstrates the triangle inequality theorem

Common pitfalls

Compare the two shorter sides to the longest. If their sum isn't greater, the arcs never meet.
Equality gives a degenerate triangle, not a real one — the points are collinear.
Keep each arc's radius set to its side length while swinging.

Frequently asked questions

How does the construction show the triangle inequality?

The arcs from the base endpoints meet only when the two shorter sides sum to more than the base, exactly the triangle inequality.

When can three lengths form a triangle?

When the two shorter lengths add to more than the longest length.

What happens when a + b = c?

The arcs just touch on the base, giving a flat, degenerate triangle with collinear points.

What does swinging an arc represent in the construction?

All the possible positions of a vertex at a fixed distance (the side length) from an endpoint of the base.

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