Real-World Applications പ്രയോഗങ്ങൾ
Carpenters, tailors, builders and map-makers use the ideas of this chapter every day, mostly without naming them. Equally spaced parallels divide things into equal parts; parallels in general divide in a ratio; midpoints halve lengths; and the centroid tells you where a triangle balances. Each tool below is one of those facts at work.
- Equally spaced parallels cut every line into equal parts (page 5).
- Three or more parallels cut any two lines in the same ratio (page 6).
- The midline of a triangle is parallel to the third side and half of it (page 8).
- The medians meet at the centroid, 2 : 1 from each vertex; the circumcentre of a right triangle is the midpoint of the hypotenuse (page 10, page 11).
1. Ruled paper divides a ribbon
You need to cut a ribbon into 7 equal pieces but 7 does not divide its length nicely. Lay it slanting across the lines of a ruled notebook so that its ends sit on line 1 and line 8, and mark the crossings. The lines are equally spaced parallels, so the pieces are equal – no measuring at all.
Change the number of parts: the ribbon re-tilts so that it spans exactly that many gaps. The steeper it is, the shorter the pieces – but they are always equal to each other.
2. The carpenter's diagonal ruler
A plank 23 cm wide must be cut into 5 equal strips. 23 ÷ 5 = 4.6 – awkward to mark. The carpenter lays the ruler diagonally across the plank so that 25 cm spans it, marks every 5 cm, and draws lines through the marks parallel to the edges. Five equal strips, guaranteed by the ratio theorem: the edges and the drawn lines are parallels cutting the ruler into equal parts, so they cut the plank's width into equal parts too.
The ruler length is chosen as the smallest convenient multiple of the number of strips that is at least the plank width. The parallels through the marks make strips of exactly width ÷ strips.
3. Folding paper into thirds
Folding a sheet in half is easy; folding it into exact thirds is not – unless you borrow the same trick.
4. Equal risers on a staircase
Ladder rungs, railway sleepers and the steps of a staircase must be equally spaced. A staircase that rises 270 cm with 15 steps needs risers of 18 cm. The builder draws the floor and landing lines, divides the rise with equally spaced parallels, and the slanted stringer is cut into equal pieces automatically.
5. Sharing land or cloth in a ratio
A 12 m strip of land must be shared between two brothers in the ratio 1 : 3 (or a piece of cloth in 2 : 3). Draw a slanted ray from one end, step off 1 + 3 equal lengths, join the last mark to the far end, and draw the one parallel needed.
6. A fence across a triangular plot
A triangular plot has one side 60 m long. A fence joins the midpoints of the other two sides. How long is the fence? By the midpoint theorem it is parallel to the 60 m side and half of it: 30 m – no need to walk across the field with a tape.
7. Where does a triangle balance?
Cut a triangle from cardboard and try to balance it on a pencil tip. It balances only at the centroid (മധ്യമകേന്ദ്രം), where the three medians meet, two-thirds of the way from each vertex. Mobiles, kites and signboards are balanced this way.
Whatever the shape, each median is cut 2 : 1 by the centroid. The pencil tip (orange) is placed exactly there.
8. The sliding ladder
A 4 m ladder slides down a wall. Its foot moves out, its top moves down – but its midpoint stays exactly 2 m from the corner all the time, tracing a circle. The ladder, the wall and the floor make a right triangle, and the midpoint of the hypotenuse is equidistant from all three vertices (it is the circumcentre).
9. Roof trusses and kite frames
In a triangular roof truss the horizontal tie joining the midpoints of the two rafters is always half the span. In a kite the two sticks (the diagonals) are perpendicular, so the frame joining the midpoints of the four edges is a rectangle – a fact builders of box kites rely on.
Each side of the inner figure is half a diagonal and parallel to it. Perpendicular diagonals ⇒ a rectangle; equal diagonals would give a rhombus; both ⇒ a square (Exercise 4, Q5).
10. Map grids and scale drawings
The grid lines on a map are equally spaced parallels. A straight road crossing them is cut into equal pieces, so counting the squares a road crosses tells you its length: each piece equals the grid spacing in real distance divided by the cosine of the tilt – or more simply, the road's length is (number of pieces) × (length of one piece).
11. Try these word problems
- Equal parts: use equally spaced parallels (ruled paper, diagonal ruler, slanted ray with equal compass steps).
- Parts in a ratio: the same construction with unequal steps – parallels cut every line in the same ratio.
- Midpoints: the midline is half the third side; the midpoint figure of any quadrilateral is a parallelogram.
- Centres: the centroid balances the triangle and cuts each median 2 : 1; a right triangle's circumcentre is the midpoint of the hypotenuse.