TrianglesSimilar Figures

From Class IX you know that two circles with equal radii are congruentsimilarequal, two squares with equal sides are congruent, and two equilateralisoscelesscalene triangles with equal sides are congruent. Equal sizes give congruence. But what can we say when the sizes are different?

The Photocopier Test

Riya helps out at her uncle's stationery shop, and its photocopier has a zoom button. Take two circles of different radii: they are not congruent, because one is bigger — and yet one is simply a zoomed copy of the other.

Fig. 6.1 — (i) circles, (ii) squares and (iii) equilateral triangles, copied at any zoom.

Some of these figures are congruent and some are not, but all of them have the same shape. So all circles are similar, all squares are similar, and all equilateral triangles are similar: similar figures have the same shape but not necessarily the same size. Congruent figures are just the special case in which the size matches too. So all congruent figures are similar, but similar figures need not be congruent.

Looking Alike Is Not Enough

Can a circle be similar to a square? Can a triangle be similar to a square? One look says no — no zoom turns a circle into a square. But what about the two quadrilaterals ABCD and PQRS in Fig. 6.2?

Fig. 6.2 — quadrilaterals ABCD and PQRS.

They look alike, but looking is not proof: a slightly stretched copy would fool your eyes. We need a precise definition of similarity, and then rules that settle the question for certain. Photographs point the way. Fig. 6.3 shows the same photograph of a monument printed in three sizes — stamp size, passport size and postcard size.

Fig. 6.3 — one photograph of a monument, printed in three sizes.

These three prints are similar: the same picture at different sizes. Now take two passport-size photographs of the same person, one at age 10 and one at age 40. They are the same size, but the face has changed shape — so the two photographs are not similarsimilarcongruent.

Some pairs of figures are similar whatever sizes you pick; others only sometimes. Drag each pair into the bucket where it belongs.

How a Photograph Grows

How does a photographer print one photograph in several sizes? She enlarges a small original — say a 35 mm film — to a bigger print, say 45 mm. Every line segment of the small picture grows in the ratio 35 : 45; its length is multiplied by . Read the other way, every segment of the big print shrinks in the ratiodifferenceproduct 45 : 35. And the angle between any two lines of the picture? It stays exactly the samegrows in the same ratioshrinks.

One ratio for every length, and no change in any angle: that is the whole idea of similarity. For polygons we say:

That common ratio of corresponding sides is called the scale factor (or the Representative Fraction). World maps and the blueprints of buildings are drawn to a chosen scale factor.

Shadow Play

Activity 1 : Fix a lighted bulb at a point O on the ceiling, with a table directly below it. Cut a quadrilateral ABCD out of flat cardboard and hold it parallel to the floor, between the bulb and the table. Its shadow falls on the table: trace the outline and call it A′B′C′D′ (see Fig. 6.4).

Fig. 6.4 — the bulb O, 240 cm above the table, the cardboard ABCD and its shadow A′B′C′D′.

The shadow A′B′C′D′ is an enlargement (or magnification) of ABCD. Light travels in straight linescurvescircles, so A′ lies on ray OA, B′ on ray OB, C′ on ray OC and D′ on ray OD: every corner is pushed away from O by the same factor. Same shape, different size — quadrilateral A′B′C′D′ is similar to quadrilateral ABCD, and ABCD is similar to A′B′C′D′.

Each corner of the shadow belongs to one corner of the card: A′ corresponds to A, B′ to B, C′ to C and D′ to D. We write these correspondences as , , and . Measure the two quadrilaterals and you find

(i) , , and , and

(ii) .

That is exactly the definition at work: all corresponding angles are equalproportionaldifferent, and all corresponding sides are in the same ratioanglesize.

So the quadrilaterals ABCD and PQRS of Fig. 6.5 are similar: matching arcs mark equal angles, and every side of PQRS is 1.4 times the matching side of ABCD.

Fig. 6.5 — PQRS is ABCD at a scale factor of 1.4.

Both Conditions, Every Time

Is one of the two conditions enough on its own? In Fig. 6.6, every angle of the square ABCD and of the rectangle PQRS is , so the corresponding angles are equal. But are the corresponding sides in one ratio?

Fig. 6.6 is live. Drag the length of PQ and read the two side ratios. Find the one length that makes them agree.

PQ = ${L} cm

Fig. 6.6 — a square and a rectangle: the angles always match, the sides need not.

The ratios agree only when PQRS stops being a rectangle and becomes a square. So a square and a rectangle that is not a square have all their angles equal, and yet they are not similarsimilar.

Now the opposite case. The square ABCD and the rhombustrapeziumkite PQRS of Fig. 6.7 both have every side 3 cm, so every side ratio is 3 : 3. Drag the angle at P. When are the two quadrilaterals similar?

${a}°

Fig. 6.7 — a square and a rhombus: the sides are in one ratio, the angles need not be equal.

Only at — when the rhombus is itself a square — are the two similar. So neither condition on its own is enough for two polygons to be similar.

Speed Round: Similar or Not?

Each card names two figures. Decide in your head: the same shape at every size — or not? After each answer you will see the reason.

Get five right to move on.