Pythagorean theorem — Mathematics, 11–13 years
In a right-angled triangle, the three side lengths follow a dependable square relationship.
The longest side and two shorter sides
A right-angled triangle has one 90° corner. The side opposite that corner is the hypotenuse, and it is always the longest. Pythagoras found that the square of its length equals the sum of the squares of the other two sides: a² + b² = c².
Why this relationship matters
Measuring a sloping distance directly can be awkward, especially when one endpoint is high or far away. The theorem turns two easier measurements, the perpendicular horizontal and vertical parts, into the missing straight distance. It was studied in ancient geometry because builders needed reliable right angles and lengths.
The 3–4–5 triangle
Suppose the shorter sides of a right triangle are 3 m and 4 m. Use a² + b² = c²: 3² + 4² = c², so 9 + 16 = 25. The square root of 25 is 5, so the hypotenuse is 5 m. Check that 5 is the longest side.
It needs a right angle
A tempting mistake is to use a² + b² = c² for every triangle. The formula works only when two sides meet at 90°, because its geometric proof depends on that corner. Without a right angle, the squares usually do not fit the relationship, even if one side looks longest.
Measuring without a straight path
A builder can use the theorem to check whether a corner is truly square by making a 3–4–5 triangle. It also finds the length of a ladder against a wall or the diagonal of a rectangular screen. These uses work when the relevant parts form a right angle.
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