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CBSE · Class 10 · Mathematics

Triangles 6.2: Similar Figures

5 concepts · 40 practice questions with answers · Free to read

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Similar and congruent figures

From daily life

A photo and its enlarged print look the same but have different sizes. A photocopy at the same size is an exact match.

Simple definition

Similar figures have the same shape, but not always the same size: corresponding angles are equal and corresponding sides are proportional. If the sizes are also equal, the figures are congruent (a special case of similar with k = 1).

smallbig (k = 2)same shape

In the figure: Same shape, different size: the big triangle is the small one enlarged twice.

Rule 1: Similar or not? — Compare shape: angles equal and sides in the same ratio.

  • Easy: All circles are similar; all squares are similar.
  • Medium: Rectangles 2 × 3 and 4 × 6: 2/4 = 3/6, so similar.
  • Exam-type: Rectangles 2 × 3 and 4 × 5: 2/4 ≠ 3/5, so not similar.

Rule 2: Congruent is a special case — Congruent figures are similar with ratio 1.

  • Easy: Two identical coins: similar and congruent.
  • Medium: A 6 cm square and a 9 cm square: similar, not congruent.
  • Exam-type: Every congruent pair is similar, but not every similar pair is congruent.
Remember

Similar = same shape; congruent = same shape and same size.

Common mistake

Calling two rectangles similar only because all angles are 90°; the sides must also be proportional.

Memory clue

"Similar: shape same. Congruent: shape and size same."

Practice questions

Easy
  1. 1Two rectangles have sides 10 cm × 5 cm and 20 cm × 10 cm. Are they similar?
  2. 2Two rectangles have sides 12 cm × 8 cm and 24 cm × 16 cm. Are they similar?
  3. 3If △BAE ~ △PQR, which angle is equal to ∠E?
Medium
  1. 4Two rectangles have sides 3 cm × 2 cm and 3 cm × 2 cm. Are they similar?
  2. 5If △WTB ~ △ZKA, which angle is equal to ∠W?
  3. 6Two rectangles have sides 3 cm × 2 cm and 9 cm × 8 cm. Are they similar?
Challenging
  1. 7If △PAK ~ △WLU, then PA/WL = AK/LU = ?
  2. 8If △NSU ~ △EZM, then NS/EZ = SU/ZM = ?
Show answers (try them yourself first)
  1. 1Yes
  2. 2Yes
  3. 3∠R
  4. 4Yes
  5. 5∠Z
  6. 6No
  7. 7KP/UW
  8. 8UN/ME

Ratio of corresponding sides

From daily life

To enlarge a 4 cm drawing to 8 cm, every length is doubled: the ratio of new to old is 2 for every side.

Simple definition

In similar figures all corresponding sides have the same ratio, the scale factor k = (side of new)/(side of old). It also equals the ratio of their perimeters.

Formula: k = new side / old side

4538106k = 2

In the figure: Sides 3, 4, 5 become 6, 8, 10: every ratio is 2.

Live figure

Ratio of corresponding sides

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Rule 1: Find the ratio k — Divide a side of the new figure by the matching side of the old one.

  • Easy: 3 cm → 6 cm: k = 2.
  • Medium: 6 cm → 9 cm: k = 9/6 = 3/2.
  • Exam-type: 4 cm → 10 cm: k = 5/2.

Rule 2: Check all sides and perimeters — Every pair must give the same ratio; perimeters follow it too.

  • Easy: 3, 4, 5 and 6, 8, 10: ratios 2, 2, 2: similar.
  • Medium: 2, 3, 4 and 4, 6, 9: ratios 2, 2, 9/4: not similar.
  • Exam-type: Sides 5, 6, 7 (perimeter 18) with k = 3: new perimeter 54.
Remember

The same k works for every pair of corresponding sides and for the perimeter.

Common mistake

Dividing old by new in one pair and new by old in another.

Memory clue

"Always new over old, same order for every side."

Practice questions

Easy
  1. 1Two similar triangles have corresponding sides 3 cm and 12 cm. What is the ratio of the first to the second?
  2. 2Two similar triangles have corresponding sides 16 cm and 24 cm. What is the ratio of the first to the second?
  3. 3Two circles have radii 12 cm and 10 cm. In what ratio are their circumferences?
Medium
  1. 4Two circles have radii 4 cm and 11 cm. In what ratio are their circumferences?
  2. 5Two similar triangles have corresponding sides 7 cm and 14 cm. What is the ratio of the first to the second?
  3. 6Two triangles are similar with corresponding sides in the ratio 4 : 3. The perimeter of the second is 33 cm. Find the perimeter of the first (in cm).
Challenging
  1. 7Two triangles are similar with corresponding sides in the ratio 1 : 4. The perimeter of the second is 148 cm. Find the perimeter of the first (in cm).
  2. 8Two triangles are similar with corresponding sides in the ratio 3 : 5. The perimeter of the second is 60 cm. Find the perimeter of the first (in cm).
Show answers (try them yourself first)
  1. 11 : 4
  2. 22 : 3
  3. 36 : 5
  4. 44 : 11
  5. 51 : 2
  6. 644
  7. 737
  8. 836

Finding unknown sides in similar figures

From daily life

If a 12 cm toy car is a model of a real car in a fixed scale, one known length gives all the others.

Simple definition

Write the equal ratios of corresponding sides, put the unknown as x, and cross-multiply; or find k first and multiply.

writeratiosputxcrossmultiplyx = value

In the figure: 3, 4, 5 ~ 6, 8, x: 5/x = 3/6, so x = 10.

Rule 1: Set up a proportion — Keep corresponding sides in the same position of each ratio.

  • Easy: 3, 4, 5 ~ 6, 8, x: x = 10.
  • Medium: 2 : 8 = 3 : y gives y = 12.
  • Exam-type: Rectangles: 4/10 = 6/x gives x = 15.

Rule 2: Use k directly — Find k once, then multiply or divide every side.

  • Easy: k = 3 (4 → 12): a side 5 becomes 15.
  • Medium: k = 2/5: a side 10 becomes 4.
  • Exam-type: k = 3/2: a side 14 becomes 21.
Remember

Check by seeing whether all ratios now match.

Common mistake

Pairing non-corresponding sides, for example the longest side with a shorter one.

Memory clue

"Same position in both ratios, then cross-multiply."

Practice questions

Easy
  1. 1△ZCT ~ △RUF. If TZ = 4 cm, FR = 20 cm and ZC = 3 cm, find RU (in cm).
  2. 2△DFS ~ △KVZ. If DF = 2 cm, KV = 10 cm and FS = 8 cm, find VZ (in cm).
  3. 3△VWZ ~ △QRU. If VW = 8 cm, QR = 24 cm and ZV = 4 cm, find UQ (in cm).
Medium
  1. 4△QFH ~ △YXC. If FH = 6 cm, XC = 9 cm and QF = 10 cm, find YX (in cm).
  2. 5△UMP ~ △FVH. If UM = 2 cm, FV = 4 cm and MP = 4 cm, find VH (in cm).
  3. 6Two rectangles are similar. The first measures 2 cm × 5 cm, and the side of the second that corresponds to the 2 cm side is 6 cm. Find the side of the second that corresponds to the 5 cm side (in cm).
Challenging
  1. 7Two rectangles are similar. The first measures 10 cm × 8 cm, and the side of the second that corresponds to the 10 cm side is 20 cm. Find the side of the second that corresponds to the 8 cm side (in cm).
  2. 8Two rectangles are similar. The first measures 9 cm × 7 cm, and the side of the second that corresponds to the 9 cm side is 27 cm. Find the side of the second that corresponds to the 7 cm side (in cm).
Show answers (try them yourself first)
  1. 115
  2. 240
  3. 312
  4. 415
  5. 58
  6. 615
  7. 716
  8. 821

Corresponding angles and vertices

From daily life

Two friends with the same photo pose match up: head with head, left hand with left hand. Triangles match vertex by vertex.

Simple definition

In ΔABC ~ ΔPQR the order of letters shows the pairs: A ↔ P, B ↔ Q, C ↔ R. So ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R and AB/PQ = BC/QR = CA/RP.

ABCPQRA↔PB↔QC↔R

In the figure: Matching letters mark matching angles; the sides between the matching letters correspond.

Rule 1: Vertex order shows the pairs — Read the letters in the same order in both triangles.

  • Easy: ΔABC ~ ΔPQR: ∠B = ∠Q.
  • Medium: ΔABC ~ ΔXYZ: side AB corresponds to XY.
  • Exam-type: AC corresponds to PR in ΔABC ~ ΔPQR.

Rule 2: Find a missing angle — Equal angles pair up; the three angles add to 180°.

  • Easy: ∠A = 50°, ∠B = 60° in ΔABC: ∠R = ∠C = 70°.
  • Medium: ΔABC ~ ΔDEF, ∠D = 40°, ∠E = 65°: ∠F = 75°.
  • Exam-type: Equal angles lie opposite corresponding sides.
Remember

The order of letters in the similarity statement is the matching map.

Common mistake

Writing ΔABC ~ ΔQPR when A matches P, so the letters no longer match.

Memory clue

"Same letter place, same angle."

Practice questions

Easy
  1. 1△STM ~ △WFN, ∠S = 46° and ∠T = 42°. Find ∠N.
  2. 2△LTA ~ △ZNV, ∠L = 47° and ∠A = 58°. Find ∠N.
  3. 3△HDK ~ △QLR, ∠D = 42° and ∠K = 79°. Find ∠Q.
Medium
  1. 4△QRL ~ △XKM, ∠Q = 46° and ∠L = 104°. Find ∠K.
  2. 5△PMT ~ △DNF, ∠P = 31° and ∠M = 60°. Find ∠F.
  3. 6△HKD ~ △CEB. In the second triangle ∠C = 34° and ∠B = 80°. Find ∠K.
Challenging
  1. 7△LWT ~ △VMA. In the second triangle ∠M = 44° and ∠A = 104°. Find ∠L.
  2. 8△VPH ~ △RSX. In the second triangle ∠R = 56° and ∠S = 65°. Find ∠H.
Show answers (try them yourself first)
  1. 192
  2. 275
  3. 359
  4. 430
  5. 589
  6. 666
  7. 732
  8. 859

Scale and enlargement applications

From daily life

Maps, models and photo prints are all similar figures: a fixed scale shrinks or enlarges every length.

Simple definition

A scale 1 : n means every length on the drawing is 1/n of the real length. Change both lengths to the same unit before using it. An enlargement by k multiplies every length by k.

Formula: real = drawing × n

real20 mscale1 : 50model40 cm

In the figure: A 20 m building at 1 : 50: 2000 cm / 50 = 40 cm model.

Rule 1: Models and maps — Convert to one unit, then multiply or divide by n.

  • Easy: 1 : 50, building 20 m: model 40 cm.
  • Medium: 1 : 50, model 6 cm: real 300 cm = 3 m.
  • Exam-type: Map 1 : 100000: 3 cm is 3 km; 8 km is 8 cm.

Rule 2: Enlargement by k — Multiply every length by k; angles stay the same.

  • Easy: Photo 4 cm × 6 cm enlarged 3 times: 12 cm × 18 cm.
  • Medium: k = 3/2 on 10 cm: 15 cm.
  • Exam-type: k = 1/4 on 8 cm: 2 cm (a reduction).
Remember

In a scale, units must match before dividing.

Common mistake

Using 1 : 50 with metres on one side and centimetres on the other.

Memory clue

"Same units first, then scale."

Practice questions

Easy
  1. 1A photograph of size 6 cm × 8 cm is enlarged (keeping its shape) so that its width becomes 24 cm. What is the new height (in cm)?
  2. 2A photograph of size 9 cm × 3 cm is enlarged (keeping its shape) so that its width becomes 45 cm. What is the new height (in cm)?
  3. 3A photograph of size 7 cm × 4 cm is enlarged (keeping its shape) so that its width becomes 14 cm. What is the new height (in cm)?
Medium
  1. 4A model of a building is made to the scale 1 : 100. The real building is 400 m high. What is the height of the model (in cm)?
  2. 5A model of a building is made to the scale 1 : 50. The real building is 135 m high. What is the height of the model (in cm)?
  3. 6On a map, 4 cm represents 10 km. Two cities are 80 km apart. How far apart are they on the map (in cm)?
Challenging
  1. 7On a map, 3 cm represents 25 km. Two cities are 375 km apart. How far apart are they on the map (in cm)?
  2. 8A model of a building is made to the scale 1 : 500. The real building is 1400 m high. What is the height of the model (in cm)?
Show answers (try them yourself first)
  1. 132
  2. 215
  3. 38
  4. 4400
  5. 5270
  6. 632
  7. 745
  8. 8280