Chapter 3 · Section 1Textbook pages 33–39

Equal Division സമഭാഗം

You have drawn parallel lines before. Now we discover something they do for us: a set of equally spaced parallel lines cuts every line that crosses them into equal parts. From this one fact comes a neat way to divide any line – 7 cm into 3 equal parts, say – using only a ruler, a compass and set squares.

📖 What you already know (from pages 1–4 of this guide)

Parallel lines (സമാന്തരവരകൾ) stay the same distance apart; a line crossing them makes equal corresponding angles with each of them.
Two right triangles with one equal side and one equal acute angle are congruent (ASA / AAS, തുല്യത്രികോണങ്ങൾ).
A compass copies a length; set squares slide to draw parallels; arcs locate the third vertex of a triangle.

1. Equally spaced parallels and a slanted line

The red lines below are parallel and equally spaced: a perpendicular (ലംബം, blue) across them is cut into equal pieces – that is what "equally spaced" means. Now tilt a line across them. Are its pieces equal too?

🖐️ Tilt the green line and measure its pieces

Tilt of the green line

The pieces of the green line are longer than the pieces of the perpendicular, but they are all equal to each other, at every tilt. The next animation shows why.

2. Why the pieces are equal

🎬 Animation: two small right triangles that must be congruent

⭐ Key idea 1 (textbook page 34)

ഒരേ അകലം ഇടവിട്ടുള്ള സമാന്തരവരകൾ മറ്റ് ഏതു വരയേയും മുറിക്കുന്നത് ഒരേ നീളമുള്ള ഭാഗങ്ങളായാണ്.

Equally spaced parallel lines cut any line into equal parts.

3. The other way round

Suppose some parallel lines cut a slanted line into equal pieces. Must the parallels be equally spaced? Draw a perpendicular and look at the same little triangles again.

🎬 Animation: equal slanted pieces ⇒ equal spacing

⭐ Key idea 2 (page 35) and Key idea 3 (page 36)

ഏതെങ്കിലും വരയെ ഒരേ നീളമുള്ള ഭാഗങ്ങളായി മുറിക്കുന്ന സമാന്തരവരകൾ ഒരേ അകലത്തിലാണ്.
Parallel lines that cut some line into equal parts are equally spaced.

Put the two ideas together: ഒരു വരയെ ഒരേ നീളമുള്ള ഭാഗങ്ങളായി മുറിക്കുന്ന സമാന്തരവരകൾ, ഏതു വരയേയും മുറിക്കുന്നത് ഒരേ നീളമുള്ള ഭാഗങ്ങളായിട്ടാണ്.
Parallel lines that cut one line into equal parts cut every line into equal parts.

🌍 Real life

Ruled notebook paper, the rungs of a ladder, railway sleepers, the steps of a staircase – all are equally spaced parallels. A carpenter who wants five equal strips from a board lays a ruler diagonally across it so that a convenient length (say 25 cm) spans the board, marks every 5 cm, and draws parallels through the marks. No division sums, no awkward decimals.

4. Dividing a 7 cm line into 3 equal parts (page 36)

7 ÷ 3 = 2.333… cannot be measured with a ruler. But 6 cm can be divided into three 2 cm pieces easily. Here is the textbook's trick.

🎬 Animation: four parallels 2 cm apart and a 7 cm compass arc

💡 Why it works

The four perpendiculars are parallel and 2 cm apart, so they cut any line – including the slanted 7 cm line – into three equal parts (Key idea 1). Each part is 7 ÷ 3 = 2⅓ cm. The lines need not even be perpendicular to the base: any four parallels spaced 2 cm along the base line will do.

5. The slanted-ray method: 10 cm into 3 equal parts (pages 37–38)

A second method needs no perpendiculars at all. Start the helping line from one end of the line to be divided.

🎬 Animation: slanted ray, equal steps, join, parallels

💡 Why it works

Imagine the three parallel lines extended, and a fourth one through A. These four parallels cut the slanted ray into equal 2 cm pieces, so (Key idea 3) they cut AB into equal pieces too: 3⅓ cm each.

6. Build it yourself: divide any line into n equal parts

🎚️ Choose the length and the number of parts

Length AB (cm) Parts n Step on the ray (cm)

The step length on the ray does not matter – only that all steps are equal. The last mark is joined to B, and parallels through the other marks do the dividing.

7. An equilateral triangle with perimeter 10 cm (page 38)

Perimeter 10 cm means each side is 10 ÷ 3 = 3⅓ cm. Divide a 10 cm line into three equal parts, then use one part as the side.

🎬 Animation: from the divided line to the triangle

A square with perimeter 10 cm is done the same way: divide 10 cm into four equal parts (2½ cm each) and draw the square on one part. Exercise 1 asks you for a triangle, a square and a hexagon like this – see page 7 of this guide for the animated solutions.

8. Ruled paper does the work for you സമാന്തര സമഭാഗം

📏 A stick across the lines of a notebook (textbook page 39)

Parts wanted

To cut a stick into 7 equal parts, lay it slanting so that one end is on line 1 and the other on line 8 (7 gaps). Mark where the lines cross it. The lines of the notebook are equally spaced parallels – Key idea 1 does the rest.

9. Check yourself

🧭 Summary of Section 1

Equally spaced parallel lines cut every line into equal parts (congruent little right triangles prove it).
Conversely, parallels that cut one line into equal parts are equally spaced, so they cut every line into equal parts.
To divide a line into n equal parts: draw a ray from one end, step off n equal lengths with the compass, join the last mark to the other end, and draw parallels through the other marks.
A triangle or square of given perimeter: divide the perimeter length into 3 or 4 equal parts first, then construct on one part.

Next: Section 2 – when the parallels are not equally spaced: division in a ratio ▶