The triangle inequality: which three lengths can close

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What the video covers, what to say before it, where a class usually goes wrong, and what to set afterwards. An account is free, and it opens every chapter of every book.

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What to assume they know

Why a compass beats trial and error for building a triangle — building a triangle from three sidelengths with two arcs, and why the crossing point is the third vertex
A circle as the points at a fixed distance from a centre; radius; two circles drawn on the same picture
Comparing sums: deciding whether one number is less than, equal to, or greater than the total of two others
The idea of supposing something is true in order to find a contradiction — used informally in Class 6 and again in The four angles at a crossing: vertically opposite and linear pairs
Reading a rough, not-to-scale diagram as a record of stated lengths

What they should be able to do

Decide, for a given triple of lengths, whether a triangle with those sidelengths exists, and justify the decision without constructing it
Explain why a straight path between two points cannot be longer than a path that detours via a third
Reconstruct the contradiction argument that rules out 10, 15 and 30
Show that only one of the three comparisons can ever fail, and identify in advance which one that is
State the triangle inequality in the chapter's own terms and apply it to a list of triples
Explain why satisfying the inequality does not, on the path argument alone, prove that the triangle exists
Relate each of the three circle cases to a comparison between the two smaller lengths and the largest
Given two lengths, describe the whole range of third lengths that would close a triangle

Where it usually goes wrong

"You have to check all three comparisons." You never do. Order the three lengths and test the largest against the other two; the other two comparisons cannot fail. The chapter walks the student to this on p.154 with a hint rather than announcing it.
"If the inequality holds, the triangle obviously exists." This is the error the chapter goes out of its way to block on p.155. The path argument is a one-way test. The circle analysis on pp.157–159 is what supplies the other direction, and it is a genuinely different argument, not a restatement.
"Equal is fine — 3, 6, 9 makes a very thin triangle." It makes no triangle. The two circles touch at exactly one point, which lies on AB itself, so the three points are in a line and there are no angles at all. This is the picture the chapter's opening question on p.146 was pointing at.
"Fig. 7.4 is drawn wrong, so the argument is unfair." It is drawn wrong on purpose. You cannot draw an impossible triangle correctly; supposing it exists and following the consequences is the only way to reach the contradiction.
"The rule is about the two shortest sides." It is about every length against the other two. The shortcut — check the longest — is a consequence, not the definition, and stating the shortcut as the rule leaves a student unable to say why it works.
"Longer sides make a triangle more likely." Scale has nothing to do with it: 10, 15, 30 fails and 10, 15, 20 succeeds; multiply any working triple by a thousand and it still works. The condition is about proportion, not size.

Questions to check understanding

Given three lengths, say whether a triangle exists and give the comparison that decides it (the printed sets on pp.154, 156 and 159 are all of this shape)
Given two lengths, state every possible third length, as a range
Explain why only one of the three comparisons has to be checked
Explain why the path argument alone cannot establish that a triangle exists
Given a triple where the two smaller lengths total exactly the largest, say what the two circles do and what the "triangle" degenerates into
Justify that an equilateral triangle exists for any positive sidelength (asked directly on p.159)
Word problems that dress the same test in distances between three places; the tent-tree-pole setting on pp.151–152 is the model for these

Examples worth working on the board

The two triples that fail by construction (Part I, §7.2, p.151). 3 cm, 4 cm, 8 cm and 2 cm, 3 cm, 6 cm. Both are set as construction tasks, so the student meets the failure with a compass in hand before any rule exists.
The tent, the tree and the pole (Part I, §7.2, pp.151–152). Checked against the printed page. The illustration on p.152 shows a palm tree at the left, a tent at the right and a thin vertical pole between and above them, with the two-leg route drawn in yellow and the direct route in red along the ground. Two children stand beside it. Inputs: three positions and two routes; the comparison is the student's.
Fig. 7.4, the rough diagram (Part I, §7.2, p.152). Checked against the printed page. It is drawn deliberately in a hand-lettered, sketchy style with the three labels written on the sides: AB = 15 cm, AC = 30 cm, BC = 10 cm. It is not to scale, and it cannot be — that is the point of calling it rough.
The three comparisons for 10, 15, 30 (Part I, §7.2, p.153). Inputs, with the arithmetic left to the explanation: BC against BA + AC; AB against AC + CB; CA against CB + BA. Two come out sensible and the third does not. The chapter's own word for the third is absurd, and the argument is a contradiction: the supposed triangle destroys itself.
The Try This observation (Part I, §7.2, p.154). For 10, 15, 30 the book prints the two comparisons that pass and the one that fails. With the lengths ordered smallest to largest, the two smaller ones automatically clear the test, because each of them is at most as big as the largest and the remaining length is a positive amount on top of that. Only the largest can fail, so there is only ever one comparison worth making.
4, 5, 8 and the admission (Part I, §7.2, p.155). The inequality holds here. The chapter then says plainly that this alone permits the triangle to exist or not to exist — it does not settle it. It is the hinge of the whole section, and an explanation that goes straight from "the test passes" to "so the triangle exists" has taught the chapter backwards.
Fig. 7.5 and the point X (Part I, §7.2, pp.155–156). Checked against the printed page. Base AB = 8 cm along a horizontal line; a hand-drawn circle of radius 4 cm centred A meets AB at a point marked X; then a circle of radius 5 cm centred B. The input the argument turns on is the length BX.
The three cases (Part I, §7.2, pp.157–159). Checked against the printed page. Three pairs of circles are drawn, all with the segment AB marked between the centres: Case 1, circles meeting at one point; Case 2, circles clear of each other; Case 3, circles overlapping. The chapter's setup for all three is printed just below the figures: AB is taken to be the longest of the three given lengths and the two radii are the smaller two. Inputs: the three pictures and that setup. How the two radii compare with AB, case by case, is what.
The four triples to test (Part I, §7.2, p.159). 1, 100, 100 · 3, 6, 9 · 1, 1, 5 · 5, 10, 12. Note that 3, 6, 9 is the borderline one: it is Case 1, not Case 2, and it is the only one of the four that sits exactly on the boundary.
The open-ended third length (Part I, §7.2, pp.159–160). Given 1 and 100, the book itself states the answer as every number strictly between 99 and 101.

Figures to have open

Two circles on a shared base AB whose radii can be shown moving continuously, so the picture passes through Case 2, Case 1 and Case 3 as the radii grow. This single step-by-step figure carries sections 11 and 12 and is the most important asset in the topic.
A ground plan with three marked positions and two coloured routes, for the path argument. Redraw; p.152's illustration is the book's own artwork.
A deliberately rough, hand-lettered triangle for Fig. 7.4. It must look provisional, not drafted.
A number-line strip for the third-length ranges, with open circles at the two endpoints to show the ends are excluded.
No table, photograph or dataset from the textbook is needed.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part I, printed Chapter 7 "A Tale of Three Intersecting Lines", §7.2 "Constructing a Triangle When its Sides are Given" — the unnumbered block "Are Triangles Possible for any Lengths?", pp.151–160
Within it: "Triangle Inequality" (bold, unnumbered), p.151 — the tent, tree and pole; Fig. 7.4 and the supposition, p.152; the three comparisons and the contradiction, p.153; the Figure it Out sets, the Try This ordering hint and the naming of the triangle inequality, p.154; the 4, 5, 8 case and the admission that the test is one-way, p.155
Within it: "Visualising the construction of circles" (bold, unnumbered), pp.155–156 — the point X and Fig. 7.5; the three cases, pp.157–159; "Conclusion" (bold, unnumbered), p.159; the closing Try This on the range of third lengths, pp.159–160
Same part, SUMMARY, p.171, second and third bullets — the chapter's own compressed statement of the inequality and of both directions of the conclusion
Backward pointer: the chapter opening, p.146, for the unanswered question about three vertices on one line