The Third Way മൂന്നാംവഴി
So far we have two ways to make a scaled copy of a triangle: copy the angles and scale one side, or scale all three sides. Is there a third way – scaling just two sides and keeping the angle between them? Yes, and it completes the story of similar triangles.
1. Two ways we already have – and a third question
Way 1 · one side + two angles
Scale one side, copy the two angles at its ends. Same angles ⇒ the other sides scale automatically (Section 1).
Way 2 · three sides
Scale all three sides and draw with a compass. Same-ratio sides ⇒ same angles automatically (Section 2).
Way 3 · two sides + the angle between?
Suppose we know two sides and the angle between them. Scale the two sides, keep the angle. Does the third side scale too? Are the other angles the same?
2. The experiment: 4 cm, 6 cm, 30° → 3 cm, 4.5 cm, 30°
Take a triangle with sides 4 cm and 6 cm and a 30° angle between them. Draw another with sides 3 cm and 4.5 cm (¾ of each) and the same 30° between them. We do not know yet whether the third side is ¾ or whether the other two angles match. Let us place one on the other.
രണ്ടു വശങ്ങളിലെ മാറ്റം ഒരേ തോതിലും, അവയുടെ ഇടയിലെ കോണുകൾ തുല്യവും ആയ ത്രികോണങ്ങളിൽ മൂന്നാം വശങ്ങളിലെ മാറ്റവും ഇതേ തോതിലാണ്. മറ്റു രണ്ടു കോണുകളും തുല്യമാണ്.
If two sides of one triangle are scaled by the same factor to give two sides of another triangle, and the angles between those sides are equal, then the third sides are scaled by the same factor too – and the other two angles are also equal.
Blue: sides 6 cm and 4 cm with the chosen angle between them. Green: sides 6k and 4k with the same angle. Whatever you choose, the third side is also k times, and the other two angles agree.
If the equal angle is not the one between the two scaled sides, the rule can fail – the triangles may have different shapes. Always check: the equal angle is included between the two sides in ratio.
3. Enlarging without measuring anything: use circles
Here is a clever use of Key idea 5. To enlarge a triangle 1½ times you do not need to measure sides or angles at all – only a compass.
Look at triangles OAB and OA′B′. They share the angle at O. OA′ = 1½ OA and OB′ = 1½ OB because the radii are 1½ times. Two sides scaled by 1½ with the same angle between ⇒ Key idea 5 ⇒ A′B′ = 1½ AB, and the angles at A and A′ are equal, so A′B′ is even parallel to AB. The same for the other two sides.
4. Other shapes: rectangles, regular polygons… but not every quadrilateral
The circle trick needs a circle through all the vertices (a circumcircle, പരിവൃത്തം). Every triangle has one. So does every rectangle and every regular polygon. A general quadrilateral – for example a parallelogram that is not a rectangle – does not.
Engineers enlarge drawings of regular parts (gears, hexagonal bolts, circular flanges) exactly this way. For an irregular shape, a pantograph (a hinged parallelogram of rods) or the "rays from a point" method of Exercise 3 does the job – no circumcircle needed.
5. Right triangles need only two pairs of sides മട്ടത്രികോണം
For two general triangles, two pairs of sides in the same ratio are not enough to say they are similar – the third pair must also be checked (or the included angle must be equal). But in a right triangle, if you know two sides, Pythagoras fixes the third. So for right triangles, two pairs of sides in the same ratio are enough.
Both legs scale by k and the angle between the legs is 90° in both – so by Key idea 5 the hypotenuse scales by k too (Pythagoras agrees). Exercise 3 asks you to prove this, and also the case "one leg and the hypotenuse".
6. What "similar" means – the complete picture
രണ്ടു ത്രികോണങ്ങൾ തമ്മിൽ ചുവടെപ്പറയുന്ന ഏതെങ്കിലും ഒരു ബന്ധമുണ്ടെങ്കിൽ, മറ്റു രണ്ടു ബന്ധങ്ങളും ഉണ്ടാകും.
If two triangles satisfy any one of the three relations below, the other two hold as well. Triangles related in this way are called similar (സദൃശം).
① Same angles
ഒരേ കോണുകളാകുക
All three angles of one triangle equal the three angles of the other. (Two are enough – the third follows from 180°.) Often called AA.
② All sides in the same ratio
വശങ്ങളിലെയെല്ലാം മാറ്റം ഒരേ തോതിലാകുക
Every side of one triangle is the same multiple k of a side of the other. Often called SSS.
③ Two sides in ratio + equal included angle
രണ്ടു വശങ്ങളിലെ മാറ്റം ഒരേ തോതിലാകുകയും, അവയുടെയിടയിൽ ഒരേ കോൺ ആയിരിക്കുകയും ചെയ്യുക
Two sides scaled by k and the angle between them equal. Often called SAS.
Compare with congruent triangles (Class 8): there the sides had to be equal; here they only need to be in the same ratio. Congruent triangles are just similar triangles with scale factor 1.
7. Projection: similar triangles from a torch പ്രക്ഷേപണം
Join the vertices of a triangle to an outside point O and extend the lines. Now draw lines parallel to the sides at any distance – you get triangles of every size, all similar to the first. This is exactly what a projector or a torch does with a shadow.
Triangles OAB and OA′B′ share the angle at O, and OA′/OA = OB′/OB = the slider value. So (Key idea 5) A′B′ is the slider value times AB – the shadow grows in proportion to its distance from the light.
സദൃശത്രികോണങ്ങളുടെ ഉയരം, നടുവരകൾ, കോൺസമഭാജികൾ എന്നിവ മാറുന്നത് വശങ്ങളുടെ അതേ തോതിലാണോ എന്ന് കണ്ടെത്തുക.
Do the heights, medians and angle bisectors of similar triangles change by the same scale factor as the sides? Think about it first – the answer, with animations, is at the end of the next page.
8. Check yourself
- Key idea 5: two sides in the same ratio + equal included angle ⇒ third sides in the same ratio and the other angles equal.
- Enlarge any triangle (or rectangle, or regular polygon) with two concentric circles – no measuring.
- Right triangles: two pairs of sides in the same ratio are enough.
- Similar triangles: same angles ⇔ all sides in the same ratio ⇔ two sides in ratio with equal included angle. Any one gives the other two.
Next: Exercise 3 – solutions with animations, plus the research task ▶