Geometry & Algebra Toolkit ജ്യാമിതിയും ബീജഗണിതവും
Section 3 of the chapter finds the areas of triangles with irrational sides and ends with Heron's formula. Everything it needs – Pythagoras, area formulas, the height of an isosceles triangle, hexagons, trapeziums, and the identities (x+y)(x−y) = x² − y² – is collected here with interactive pictures.
1. Pythagoras' theorem പൈതഗോറസ് തത്വം
In a right triangle with perpendicular sides a and b and hypotenuse c (the side opposite the right angle): a² + b² = c². Knowing any two sides gives the third – and that third side is often a square root: legs 1 and 1 give c = √2, legs 1 and √2 give c = √3.
Orange dots can be dragged (A along the floor, B up the wall). Lengths are in units of 60 px.
2. Area of a rectangle: length × breadth ചതുരത്തിന്റെ പരപ്പളവ്
With whole-number sides you simply count squares. With a side like 2.5 the half-squares still add up to length × breadth. The chapter asks: what if a side is √3, which is not even a fraction? (Answer: squeeze with decimals – Section 1.)
3. Area of a triangle: ½ × base × height
Area = ½ × base × height, where the height (ഉയരം) is the perpendicular distance from the opposite vertex to the base – measured straight down, never along a slanted side. Finding that height is the whole game in Section 3.
4. The height of an isosceles triangle bisects the base
In an isosceles triangle (two equal sides, സമപാർശ്വത്രികോണം) the perpendicular from the apex lands exactly on the midpoint of the base. This is why the half-base appears in every height calculation of the chapter.
5. A regular hexagon is six equilateral triangles സമഷഡ്ഭുജം
6. Area of a trapezium ലംബകം
Area = ½ × (sum of the parallel sides) × height. Exercise 1 of the chapter builds a trapezium from a square and an equilateral triangle – you will need this.
7. Algebra: the identities we need സർവസമവാക്യങ്ങൾ
These Class 8 identities hold for all numbers – including irrational ones like √2. The chapter uses them to prove (√2 + 1)(√2 − 1) = 1 and to derive Heron's formula.
(x − y)² = x² − 2xy + y². Check with numbers: (5 − 2)² = 9 and 25 − 20 + 4 = 9 ✓. Check with an irrational: (√2 − 1)² = 2 − 2√2 + 1 = 3 − 2√2 ≈ 0.172, and (1.414 − 1)² = 0.414² ≈ 0.171 ✓.
Whenever you see "something squared minus something squared", write it as (sum)(difference). The textbook uses exactly this with x² − (8 − x)² = (x + 8 − x)(x − 8 + x) = 8(2x − 8).
8. Check yourself
- Pythagoras: a² + b² = c²; legs 1, 1 → √2; legs 1, √2 → √3; legs 3, 4 → 5.
- Rectangle: length × breadth. Triangle: ½ × base × height. Trapezium: ½ × (sum of parallel sides) × height.
- The height of an isosceles (or equilateral) triangle meets the base at its midpoint.
- Regular hexagon of side a = 6 equilateral triangles of side a.
- (x+y)(x−y) = x² − y², (x+y)² = x² + 2xy + y², (x−y)² = x² − 2xy + y² – true for irrational numbers too.