Triangle Inequality Theorem - Interactive Activity

Three lengths do not always make a triangle. The two shorter pieces have to be long enough to reach each other when they are hinged to the ends of the longest piece. These activities let you test sets of lengths, find the range for a missing side, and work through the test one step at a time.

Definitions and Main Ideas

Not every set of three lengths can form a triangle. The two shorter pieces have to be able to reach each other when they are hinged to the ends of the longest piece — if their sum is less than the longest piece they fall short, and if their sum is exactly equal they lie flat on top of it instead of popping up into a triangle. The Triangle Inequality Theorem turns that picture into a test, and the same test produces the possible range for a third side when the other two sides are known.

Vocabulary

Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side; for three given lengths, if the sum of the two smaller lengths is greater than the longest length, the measurements will form a triangle.
Third Side Range: Given two sides of a triangle, the third side must measure more than the difference of the two sides and less than their sum.
Segment: A part of a line that consists of two endpoints and all the points between them; segments can only meet at their endpoints, which is why three lengths must be long enough to reach one another to form a triangle.
Third Side: The remaining side of a triangle when the other two sides are given; its possible lengths are exactly what the Triangle Inequality Theorem restricts.

Main Ideas

To test three lengths, add the two smaller ones: the sum must be strictly greater than the longest length.
Equality fails — if the two smaller lengths add to exactly the longest length, the pieces lie flat and no triangle forms.
Given two sides a and b, the third side x must satisfy (larger − smaller) < x < (larger + smaller).
Segments can only meet at their endpoints, which is why the two shorter pieces must be long enough to reach each other and pop up into a triangle.

Can It Form? Explorer

What to do: Move the three sliders to set three lengths. Watch the two shorter pieces swing up from the ends of the longest piece. Try to make them meet: do they pop up into a triangle, fall short, or lie flat?

Side 1 7 Side 2 10 Side 3 13

The sliders set the three lengths. The longest length is always drawn as the base, and the two shorter pieces are hinged to its ends. Watch the readout: it compares the sum of the two smaller lengths with the longest length.

Third Side Range

What to do: Pick a pair of known sides, then slide the candidate third side. Watch its color: green inside the range, amber exactly at an end, red outside. Then use Check a value to test named candidates one at a time.

The preset buttons choose the two known sides. The slider below chooses the candidate third side, and the diagram redraws it in its verdict color.

Candidate third side 15

Check a value

Look at the three segments. Can they form a triangle? Choose Yes or No. If your first choice is not correct, you will get a hint and another try.

The Next check button cycles through all ten checks in order.

Build the Test

What to do: Work each problem one step at a time. First choose the statement the problem calls for, then substitute, then solve, then check. The diagram shows the three lengths as separate segments until the final step is answered.

Choose an answer for the current step. Continue moves to the next step once the step is answered. New Problem moves to the next problem in order.

Your Turn

What to do: Answer each question. A first miss gives a hint and another try; a second miss shows the full explanation. Your score counts first-try answers only.

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Triangle Inequality Theorem — Mr. J Geometry. All activities on this page run entirely in your browser.