Pythagoras' theorem and its converse
Build squares on the sides of a right-angled triangle and compare their areas, see why the theorem is true with four copies of the triangle, find a missing side, use the converse to test for a right angle, and mark √2 on the number line with compasses.
- UK: Year 9
- France: grade 8
- Germany: grade 8
- Iran: grade 8
- Afghanistan: grade 8
- Syria: grade 8
- Palestine: grade 6
- Ukraine: grade 8
- Russia: grade 8
Guide
This studio is for Year 9 (KS3 geometry: Pythagoras' theorem). The Iranian books teach it in Year 8 «رابطهٔ فیثاغورس», the Afghan books in grade 8.
In a right-angled triangle the side opposite the right angle is the hypotenuse, and it is always the longest side. Build a square on each side: the area of the square on the hypotenuse equals the sum of the areas of the other two squares. With legs a and b and hypotenuse c: a2 + b2 = c2.
Careful: the sides themselves do not add up. The hypotenuse is shorter than the other two sides together (3 + 4 ≠ 5), but the squares do: 9 + 16 = 25. To find the hypotenuse, add the squares and then take the square root; to find a shorter side, subtract its square from the square of the hypotenuse.
The converse is true too: if the square of the longest side equals the sum of the squares of the other two, the triangle is right-angled, and the right angle is opposite the longest side. So you can test for a right angle without a protractor.