Unit 3 · Topic 03 · Geometry and Triangles
Kabir wanted to know the diagonal length of a rectangular field, 3 m by 4 m, without walking across it.
Zara said one right-angled triangle and one theorem would do it.
The answer came out to be a suspiciously round number.
Kabir wanted the diagonal distance across a rectangular field measuring 3 m by 4 m, to know the shortest path from one corner to the opposite one. Zara pointed out that the two sides of the field, together with the diagonal, formed a right-angled triangle — the diagonal was the hypotenuse, the side opposite the right angle, and always the LONGEST side of a right triangle.
She wrote the Pythagoras theorem: in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. hypotenuse² = base² + height². For the field: hypotenuse² = 3² + 4² = 9 + 16 = 25, so hypotenuse = √25 = 5 m.
"That's suspiciously neat," Kabir said. Zara explained that 3-4-5 is a well-known 'Pythagorean triple' — a set of three whole numbers that satisfy the theorem exactly. Other common triples include 5-12-13 and 8-15-17, useful for spotting quick answers without a calculator.
Kabir then flipped the problem: given the hypotenuse (13) and one side (5), find the other side. He rearranged the formula: height² = hypotenuse² − base², so height² = 13² − 5² = 169 − 25 = 144, giving height = 12. "So the theorem works in reverse too," he said, "as long as I know the hypotenuse and one other side."
Zara then showed the converse use: given a triangle with sides 6, 8, and 10, check whether it's right-angled. 6² + 8² = 36 + 64 = 100, and 10² = 100. Since they match, the triangle MUST be right-angled — this is the CONVERSE of the Pythagoras theorem, used to test for a right angle without measuring it directly.
"So the theorem does two jobs," Kabir summarised. "Given a right triangle, find a missing side. Given all three sides, check whether it's actually right-angled at all — using exactly the same equation, just applied in different directions."
Pythagoras Theorem: in a right-angled triangle, hypotenuse² = base² + height² (or a² + b² = c², where c is the hypotenuse, the longest side, opposite the right angle).
The theorem can be rearranged to find any missing side: hypotenuse = √(base² + height²), or base = √(hypotenuse² − height²), or height = √(hypotenuse² − base²).
A Pythagorean triple is a set of three whole numbers satisfying the theorem exactly, like 3-4-5, 5-12-13, and 8-15-17 — worth recognising to save calculation time.
The CONVERSE of the theorem tests whether a triangle is right-angled: if a² + b² = c² for the three sides (with c the longest), the triangle must have a right angle opposite side c.
A ladder 13 m long leans against a wall, its base 5 m from the wall. Find how high up the wall it reaches.
Check whether a triangle with sides 7, 24, and 25 is right-angled, using the converse of the Pythagoras theorem.
Loading questions…