8.2 The Triangle Inequality, the Pythagorean Theorem & Its Converse

8.2 The Triangle Inequality, the Pythagorean Theorem & Its Converse

Skill 7 of Competency 3 names three theorems explicitly, and the converse is named separately from the theorem itself — a signal that classification items appear, not just side-length computations.

The Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

For sides a, b, c all three inequalities must hold: a + b > c, a + c > b, and b + c > a. In practice, checking only the two shortest against the longest is sufficient — if the two smallest sum to more than the largest, the other two conditions follow automatically.

Can 4, 9, and 14 form a triangle? 4 + 9 = 13, which is not greater than 14. No. Can 6, 8, and 11 form a triangle? 6 + 8 = 14 > 11 ✓, 6 + 11 = 17 > 8 ✓, 8 + 11 = 19 > 6 ✓. Yes.

Note the strictness: 4 + 9 = 13 < 14 fails, and a case like 5 + 7 = 12 with a third side of exactly 12 also fails, because the three points would be collinear — a degenerate "flat" triangle, not a triangle.

The range for a third side

Given two sides a and b, the third side x must satisfy

|a − b| < x < a + b

With sides 7 and 12, the third side satisfies 5 < x < 19. Integer possibilities run from 6 to 18.

This range formulation is the most frequently tested version of the theorem. Items ask for the number of possible integer lengths — here 18 − 6 + 1 = 13 values — or which given value is impossible.

A related ordering rule: in any triangle, the longest side lies opposite the largest angle, and the shortest side opposite the smallest angle. In a triangle with angles 40°, 60°, and 80°, the side opposite the 80° angle is longest.

The Pythagorean Theorem

For a right triangle with legs a and b and hypotenuse c: a² + b² = c²

The hypotenuse is always the side opposite the right angle and is always the longest side. Assigning a leg's length to c is the single most common error.

Legs 9 and 12: c² = 81 + 144 = 225, so c = 15. Hypotenuse 26, one leg 10: 10² + b² = 26² → b² = 676 − 100 = 576 → b = 24.

The second case is where errors cluster. When the hypotenuse is known, you subtract; adding gives a nonsense result longer than the hypotenuse.

Pythagorean triples

Memorizing these saves radical arithmetic:

+---------------------------------------------------------------------------+
|  3-4-5       and multiples: 6-8-10, 9-12-15, 12-16-20, 15-20-25           |
|  5-12-13     and multiples: 10-24-26, 15-36-39                            |
|  8-15-17     and multiples: 16-30-34                                      |
|  7-24-25     and multiples: 14-48-50                                      |
|  9-40-41                                                                  |
+---------------------------------------------------------------------------+

Spotting 10-24-26 as twice 5-12-13 answers the earlier example instantly.

Special right triangles

A square with side 6 has diagonal 6√2 ≈ 8.49. An equilateral triangle with side 10 has height 5√3 ≈ 8.66, since the altitude creates two 30-60-90 triangles with short leg 5.

The second result is worth storing: the height of an equilateral triangle with side s is (s√3)/2, and its area is (s²√3)/4.

The converse and triangle classification

Converse of the Pythagorean Theorem: if a² + b² = c² for the three sides of a triangle with c longest, then the triangle is a right triangle.

The comparison extends into a full classification test. Let c be the longest side:

+---------------------------------------------------------------------------+
|   a^2 + b^2 = c^2   ->  RIGHT triangle                                    |
|   a^2 + b^2 > c^2   ->  ACUTE triangle    (all angles < 90)               |
|   a^2 + b^2 < c^2   ->  OBTUSE triangle   (one angle > 90)                |
+---------------------------------------------------------------------------+

Sides 7, 9, 12: longest is 12. Compare 49 + 81 = 130 against 144. Since 130 < 144, the triangle is obtuse. Sides 8, 10, 12: compare 64 + 100 = 164 against 144. Since 164 > 144, the triangle is acute. Sides 9, 12, 15: 81 + 144 = 225 = 15². Right.

The intuition is that as the angle opposite c opens wider, c grows relative to the legs. A larger c² means a wider angle, hence obtuse.

Identifying the longest side first is mandatory. Comparing the wrong pair reverses the conclusion, and items order the sides irregularly to test exactly this.

Applied problems

Right triangle reasoning shows up as distance, height, and diagonal questions.

A 17-foot ladder leans against a wall with its base 8 feet from the wall. How high does it reach? The ladder is the hypotenuse: 8² + h² = 17² → h² = 289 − 64 = 225 → h = 15 feet.

A rectangular field is 120 m by 50 m. How much shorter is the diagonal path than walking two sides? Diagonal = √(14,400 + 2,500) = √16,900 = 130 m. Two sides = 170 m. The shortcut saves 40 m.

A television's screen is measured on the diagonal. A 16 : 9 screen has a 55-inch diagonal. Its width and height satisfy (16k)² + (9k)² = 55², so 337k² = 3,025 and k ≈ 2.996, giving width ≈ 47.9 in and height ≈ 27.0 in.

Three-dimensional versions extend the theorem: the space diagonal of a rectangular box with edges l, w, h is √(l² + w² + h²). A 3 × 4 × 12 box has space diagonal √(9 + 16 + 144) = √169 = 13.