Pythagoras and surds: a brace for the gate
Cut a diagonal brace for a wooden gate: square the sides, add them, and find out when the brace is a whole number and when it is a surd.
Core concept
In a right-angled triangle a² + b² = c²; when c² is not a perfect square, c is an irrational surd that may simplify to k√m
Topic: Pythagoras' theorem and surds
A carpenter makes a rectangular wooden gate and nails a diagonal brace across it so it does not sag. The bottom, the side and the brace form a right-angled triangle. Pythagoras' theorem says the square on the longest side equals the squares on the other two: a² + b² = c². So a 3 m by 4 m gate needs a brace of √(9 + 16) = √25 = 5 m. For a 1 m by 1 m gate the brace is √2 m, a number that never ends or repeats as a decimal: A number with a point, like 2.5, where the digits after the point are tenths, hundredths … All key words: an irrational number, written as a surd. Some surds simplify: √8 = √(4 × 2) = 2√2.
What you will learn
- Use a² + b² = c² to find the hypotenuse of a right-angled triangle.
- Tell rational lengths from irrational ones (surds).
- Simplify surds such as √8 = 2√2 and √18 = 3√2.
Before you start
- Squares and square roots.
Materials
- Sticks or string and a set square
Equipment
- A measuring tape
Procedure
- Tap − or + to change the width and the height of the gate.
- Count the tiles in the squares on the two short sides, and compare with the square on the brace.
- Record a 3 m by 4 m gate and a 1 m by 1 m gate.
- Record a gate whose brace simplifies to k√m, and one whose surd cannot be simplified.
How this simulation is simplified
- The gate is a rectangle, so its corners are right angles.