Scale and Approximation തോത്
Two ideas power this chapter. First, scale: enlarging a figure multiplies every length by the same number. Second, approximation: a list of numbers can come closer and closer to a value without ever reaching it – the way 1.4, 1.41, 1.414 … close in on √2.
- Similar triangles (സദൃശത്രികോണങ്ങൾ): same angles, sides in the same ratio (chapter 6).
- √2 = 1.41421… is not a fraction; its decimals go on for ever (chapter 2, പുതിയ സംഖ്യകൾ).
1. Scaling a figure scales every length
If a figure is scaled by a factor k, every length in it – sides, perimeter, heights – is multiplied by k, while every area is multiplied by k².
2. Scaling a polygon in a circle
3. Closer and closer
Some lists of numbers come closer and closer (അടുത്തടുത്തു വരുന്നു) to a fixed value. The gap gets as small as we like, though it may never be exactly zero.
The perimeters of regular polygons inside a circle, with 3, 4, 5, 6, … sides, come closer and closer to the circumference. If two such lists are always in the ratio 1 : 2, the numbers they approach must also be in the ratio 1 : 2. That is how the chapter proves that doubling the diameter doubles the circumference.
4. Fractions, decimals and rounding
Rounding rule: look at the next digit; if it is 5 or more, add 1 to the last digit kept. In this chapter π ≈ 3.14 (2 places) or 3.1416 (4 places).
A photocopier set to 150% makes every length 1.5 times – and the paper covered 2.25 times. Maps and model cars use scale. Calculators show rounded values: no screen can show all the digits of √2 or π.
5. Check yourself
- Scaling by k multiplies every length by k and every area by k².
- A regular polygon in a circle of k times the radius has every side, and its perimeter, k times as long.
- A sequence can come closer and closer to a value; polygon perimeters approach the circumference.
- Round with the next-digit rule; π ≈ 3.14 or 3.1416.