6 min readConcept sheet1-mark MCQ and assertion-reason; sometimes a 2-mark 'give a reason' question

Similar figures and similar triangles

Figures are similar when they have the same shape, but maybe not the same size. Polygons are similar when angles match and sides are in one ratio.

Similar polygons (same number of sides)Corresponding angles equal AND corresponding sides in the same ratio

Always similarAll circles, all squares, all equilateral triangles

Symbol~ means 'is similar to', e.g. ΔABC ~ ΔPQR

Congruent vs similarCongruent figures are always similar; similar figures need not be congruent

What similar means

Two figures are similar if they have the same shape, but not always the same size. A photo and its enlargement are similar. A map of India and the real outline of India are similar in shape. Congruent figures have the same shape and the same size. So congruent figures are always similar, but similar figures need not be congruent.

Both conditions are needed: two counter-examples

Square and rhombusTake a square of side 3 cm and a rhombus of side 3 cm with angles 60° and 120°. Their sides are in the same ratio (1 : 1). But their angles are not equal, so they are not similar.

Square and rectangleA square and a rectangle that is not a square have all angles equal (90°). But their sides are not in one ratio, so they are not similar.

For polygons with four or more sides, check both conditions. For triangles, one condition brings the other with it (see the criteria topic).

Similar triangles

Two triangles are similar if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio (proportional). If ΔABC ~ ΔPQR, then ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R and AB/PQ = BC/QR = CA/RP. The order of the letters matters: A matches P, B matches Q, C matches R. The common ratio is called the scale factor.

Example 1: a missing side

Problem

ΔABC ~ ΔDEF. AB = 4 cm, BC = 6 cm, CA = 5 cm and DE = 6 cm. Find EF and FD.

Working

ΔABC ~ ΔDEF, so AB/DE = BC/EF = CA/FD
4/6 = 6/EF, so EF = (6 × 6)/4 = 36/4 = 9 cm
4/6 = 5/FD, so FD = (5 × 6)/4 = 30/4 = 7.5 cm

Answer

EF = 9 cm and FD = 7.5 cm

Example 2: enlarging a photo

Problem

A passport photo is 3.5 cm wide and 4.5 cm tall. A studio enlarges it so that the width becomes 14 cm and the shape stays similar. What is the new height?

Working

Similar rectangles have their sides in the same ratio.
Scale factor = new width/old width = 14/3.5 = 4
New height = 4.5 × 4 = 18 cm

Answer

The new height is 18 cm.

In real life

A map of India in your atlas has the same shape as the real outline of India. A studio enlargement of a photo and a scale model of the Red Fort are similar to the originals too.

Exam angle

1 marks: “If ΔABC ~ ΔQRP, which angle of ΔQRP is equal to ∠B?” Match the letters in order: A with Q, B with R, C with P. So ∠B = ∠R.

2 marks: “Are all squares similar? Are all rectangles similar? Give reasons.” All squares: yes, all angles are 90° and all sides are in one ratio. All rectangles: no, the angles match but the sides need not be in one ratio.

2 marks: “Are a square of side 4 cm and a rhombus of side 4 cm always similar? Give a reason.” No. The sides are in the same ratio, but the angles of the rhombus need not be 90°.

Remember it

Same shape, any size = similar. Same shape, same size = congruent. For polygons, check angles AND ratios.