Right Triangles, Pythagoras & Squares on the Sides മട്ടത്രികോണവും പൈഥഗോറസ് തത്വവും
Every new number in this chapter is born inside a right triangle: the diagonal of a square, the height of an equilateral triangle, the hypotenuse of a triangle with sides 1 and √2. The tool that links the sides is the theorem of Pythagoras, stated here the way the textbook uses it – with squares drawn on the sides.
- A right angle is 90°; the small square mark ∟ shows it.
- Area of a square = side × side (page 1); the three angles of a triangle add up to 180°.
- Two triangles are congruent (തുല്യം) when they match exactly – for example two right triangles with the same two perpendicular sides.
1. The parts of a right triangle
The side opposite the right angle is the hypotenuse (കർണ്ണം) – always the longest side. The two sides that make the right angle are the perpendicular sides (ലംബവശങ്ങൾ).
2. Pythagoras with squares on the sides
Draw a square on each side of a right triangle. The theorem of Pythagoras says: the square on the hypotenuse has the same area as the two squares on the perpendicular sides together. For sides 3, 4, 5 you can count the unit squares.
In a right triangle with perpendicular sides a, b and hypotenuse c: a² + b² = c² – the area of the square on c equals the areas of the squares on a and b added together.
3. The chapter's opening picture: sides 1 and 1
When both perpendicular sides are 1, the square on the hypotenuse can be cut into four triangles – each half of a unit square. So its area is 2 = 1 + 1, exactly as Pythagoras promises.
4. Finding a side when two are known
When the hypotenuse is known, subtract the squares: the square on the missing side = square on the hypotenuse − square on the known side. The chapter's examples: hypotenuse 2 and side 1 give a square of area 3; sides 1 and √2 give a hypotenuse square of area 1 + 2 = 3.
5. The height of an equilateral triangle cuts the base in half
6. One side is always shorter than the other two together
This is why the diagonal of the unit square is less than 2: it is one side of a triangle whose other two sides are 1 and 1.
However you drag C, the two slanted sides together are longer than the base – the straight path from A to B is the shortest. Push C down towards the base and the sum gets close to the base, but it reaches it only when the triangle collapses flat.
7. Drawing them with ruler and compass
8. The fan of right triangles: √2, √3, √4, √5, …
Start with perpendicular sides 1 and 1. Use the hypotenuse as a side of the next triangle, with a new side 1 at right angles, and keep going. The textbook draws this fan on pages 26 and 30.
The nth triangle has sides 1, √n and √(n + 1): each new hypotenuse-square is the old one plus 1. Every third, eighth, fifteenth… hypotenuse is a whole number (√4 = 2, √9 = 3, √16 = 4).
9. Check yourself
- Hypotenuse (കർണ്ണം) opposite the right angle; perpendicular sides (ലംബവശങ്ങൾ) make the right angle.
- Pythagoras: square on the hypotenuse = sum of the squares on the perpendicular sides. 3-4-5: 9 + 16 = 25; 1-1: 1 + 1 = 2.
- Missing perpendicular side: subtract the squares (hypotenuse 2, side 1 → square of area 3).
- The height of an equilateral triangle bisects the base; any side of a triangle is shorter than the other two together.
- The fan of unit right triangles produces hypotenuses √2, √3, √4 = 2, √5, …
Next: Odd and even numbers, lowest terms and the square identities ▶