Chapter 6 · Section 2Textbook pages 107–112

Sides and Angles വശങ്ങളും കോണുകളും

Section 1 went from angles to sides: same angles ⇒ sides in the same ratio. Now we turn it around. If every side of a triangle is multiplied by the same number, do the angles stay the same? And what happens to the area?

📖 Tools from earlier pages

Converse of the parallel-line theorem (page 3): a line that cuts two sides of a triangle in the same ratio is parallel to the third side.
SSS congruence (page 2): two triangles with the same three side lengths are equal in every way – including their angles.
Key ideas 1 and 2 (page 5): same angles ⇒ sides in the same ratio.

1. The question

Here are two triangles. The small one has sides 2, 4, 5 cm. The big one has sides 3, 6, 7.5 cm – every side is exactly 1½ times. We did not measure any angles. Are the angles the same anyway?

🎬 Animation: the sides are scaled – so the angles must match

💡 The shape of the argument

We built a copy of the small triangle inside the big one (using a parallel line), showed the copy has the big triangle's angles, and then showed the copy is exactly the small triangle. Three links in a chain – each one from an earlier page.

2. The same proof for any scale factor k

The numbers 2, 4, 5 played no special role. Let the small triangle have sides a, b, c and the big one ka, kb, kc. Move the slider to change k – the argument never changes.

🎬 General proof (with a slider for k)

⭐ Key idea 3 (textbook page 110)

രണ്ടു ത്രികോണങ്ങളുടെ വശങ്ങളുടെയെല്ലാം നീളത്തിന്റെ മാറ്റം ഒരേ തോതിലാണെങ്കിൽ, അവയ്ക്ക് ഒരേ കോണുകളാണ്.

If all the sides of two triangles change by the same scale factor, the two triangles have the same angles.

3. Scaling a triangle without a protractor

This is the practical pay-off. To make a bigger or smaller copy of a triangle you never need to measure its angles: just multiply the three sides by the same number and draw the new triangle with a compass (SSS construction). The angles come out right by themselves.

🎚️ Scale 4-6-8 by any factor

scale factor k

Only the side lengths were multiplied. The angle arcs stay identical – that is Key idea 3 in action.

🌍 Real life

A carpenter making a larger roof bracket, a tailor grading a pattern to a bigger size, or a student drawing a scale model of a triangular garden – all of them just multiply the side lengths. No protractor, no trigonometry. The shape takes care of itself.

4. What happens to the area?

If every side is multiplied by r, is the area also multiplied by r? Let's find out with the formula area = ½ × base × height.

🎬 Animation: area scales by r²

⭐ Key idea 4 (textbook page 111)

പരപ്പളവ് മാറുന്ന തോത്, വശങ്ങൾ മാറുന്ന തോതിന്റെ വർഗമാണ്.

When the sides of a triangle are multiplied by r, the area is multiplied by r².

🖐️ See it: how many small triangles fit inside the big one?

sides × r

Sides doubled ⇒ 4 copies fit; sides tripled ⇒ 9 copies. The area grows with the square of the scale factor, because area is "length × length".

Quick table

Sides multiplied by r	½	¾	1	1½	2	3	4	10
Area multiplied by r²	¼	9/16	1	2¼	4	9	16	100

Going the other way: if the area is multiplied by 25, the sides are multiplied by √25 = 5.

🌍 Real life: pizzas, prints and maps

Pizza: a 12-inch pizza and an 8-inch pizza are the same shape. Scale factor 12/8 = 1.5, so the big one has 1.5² = 2.25 times as much pizza – more than two small ones!
Photo prints: a 4 × 6 print enlarged to 8 × 12 has sides × 2 and needs 4 times the paper.
Maps: on a 1 : 50 000 map, lengths are 50 000 times smaller, but areas are 50 000² = 2 500 000 000 times smaller.

5. Check yourself

🧭 Summary of Section 2

Key idea 3: all sides × the same k ⇒ same angles. (Proof: build a copy inside the big triangle with a parallel line, then use SSS.)
So Section 1 and Section 2 together say: same angles and sides in the same ratio always come together.
To scale a triangle, multiply the three sides – no angles needed.
Key idea 4: sides × r ⇒ area × r².

Next: Exercise 2 – solutions ▶