Division in a Ratio അസമഭാഗങ്ങൾ
In Section 1 the parallel lines were equally spaced and every crossing line was cut into equal parts. What if the spacing is not equal – say 2 cm and 3 cm? The pieces are no longer equal, but they keep a fixed ratio (അംശബന്ധം). This lets us divide a line in any ratio we like, and build rectangles and triangles with a given perimeter and given side ratios.
- Equally spaced parallels cut every line into equal parts (Section 1, Key ideas 1–3).
- Dividing a length L in the ratio m : n gives pieces L × m/(m + n) and L × n/(m + n) (page 3).
- Triangles with the same height have areas in the ratio of their bases; triangles on the same base with apexes on a line parallel to the base have equal areas (page 3).
- Set squares draw parallels; a compass copies lengths and marks equal steps (page 4).
1. Unequal spacing: 2 cm and 3 cm
Three parallel lines, the lower gap 2 cm and the upper gap 3 cm. A slanted line across them is cut into two pieces. They are not 2 cm and 3 cm – so are they related to 2 and 3 at all?
The two pieces change length as you tilt, but the lower piece is always 2 parts and the upper piece 3 parts of the same unit: their ratio stays 2 : 3, the ratio of the gaps.
2. The general question: a : b = p : q?
Take any three parallel lines and two lines crossing them. The left line is cut into pieces a and b, the right line into p and q. The gaps now may be anything – the parallels need not be "nice". Is it still true that a : b = p : q, that is, ab = pq?
Drag the ends along the top and bottom lines. However the two crossing lines lean, a/b and p/q show the same number – it is fixed by the parallels alone (it equals the ratio of the two gaps).
3. Why: turn lengths into areas
Measuring convinces, but a proof explains. The textbook's idea is clever: compare the lengths through the areas of triangles.
- Two triangles with the same apex and bases on the same line have the same height, so their areas are in the ratio of their bases: a : b = A : B.
- Two triangles on the same base whose third vertices lie on a line parallel to the base have the same height, so they have equal areas: A = P.
More than three parallels
മൂന്നോ അതിലധികമോ സമാന്തരവരകൾ, ഏതു രണ്ടു വരകളെയും മുറിക്കുന്നത് ഒരേ അംശബന്ധത്തിലാണ്.
Three or more parallel lines cut any two lines in the same ratio.
4. Dividing a line in a ratio: 8 cm in 1 : 2 (page 43)
8 cm in the ratio 1 : 2 means pieces of 8 × ⅓ = 2⅔ cm and 8 × ⅔ = 5⅓ cm – again lengths you cannot read off a ruler. A 6 cm line is easy to divide 1 : 2 (2 cm and 4 cm), so let parallels transfer the ratio.
On the ray, step off m units and then n units (any convenient unit). Join the last mark to B; the parallel through the first mark cuts AB in the ratio m : n. Three parallels – the join, the parallel, and the "parallel through A" you can imagine – cut the ray and AB in the same ratio.
5. A rectangle with perimeter 30 cm and sides 5 : 3 (page 44)
Perimeter 30 cm means length + breadth = 15 cm. So we need to divide a 15 cm line in the ratio 5 : 3; the two pieces are the length and the breadth.
6. A triangle with perimeter 10 cm and sides 2 : 3 : 4 (page 45)
Straighten the three sides of the triangle into one line: its length is the perimeter, 10 cm. Divide that line in the ratio 2 : 3 : 4 and the three pieces are the three sides.
Any one. Keep one piece on the line as a side; open the compass to each of the other two pieces and draw arcs from the ends of the base; the arcs meet at the third vertex. The textbook keeps the first piece as the base.
A tailor cutting a 1.3 m strip of cloth so that two children get pieces in the ratio 3 : 4, a farmer splitting a field between two sons in the ratio of their shares, a cook dividing a rolled-out dough strip 2 : 3 – all can be done with a slanted line and parallels, no decimals needed. Map makers use the same idea in reverse: a scale bar divided in equal parts is made with a diagonal "scale" drawn across parallel lines.
7. Check yourself
- Parallels with unequal gaps cut a crossing line into unequal pieces, but the pieces are in a fixed ratio (equal to the ratio of the gaps).
- Three or more parallel lines cut any two lines in the same ratio – proved by comparing areas of triangles.
- To divide AB in the ratio m : n: ray from A, m steps then n steps, join the last mark to B, draw the parallel through the m-th mark.
- Rectangle with perimeter P and sides m : n: divide a line of length P/2 in m : n. Triangle with perimeter P and sides l : m : n: divide a line of length P in l : m : n, then swing arcs.