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About Similarity — Class 10 ICSE

Understand similarity criteria for triangles and apply BPT (Basic Proportionality Theorem). This topic is part of the ICSE Class 10 mathematics syllabus (chapter: Chapter 4). On this page you can practice 47 questions across three difficulty levels — 20 easy, 19 medium, and 8 hard — each with a visual step-by-step solution, plus a timed 24-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

What you'll learn in Similarity

  • Introduction to Similarity: What Makes Shapes 'Look Alike'?
  • Criteria for Similarity of Triangles: Proving Shapes are Alike
  • Basic Proportionality Theorem (BPT) / Thales Theorem
  • Advanced Applications: Areas of Similar Triangles
  • Mastering Similarity: Connecting Concepts and Exam Preparation

Interactive lesson · about 15 minutes · checkpoint question after every unit

Similarity — solved examples for Class 10 ICSE

Example 1easy

Which of the following statements correctly distinguishes between similar and congruent figures?
  1. A)A. Similar figures have the same size and same shape, while congruent figures have the same shape but different sizes.
  2. B)B. Similar figures have the same shape but not necessarily the same size, while congruent figures have both the same shape and the same size.
  3. C)C. Similar figures have different shapes but the same size, while congruent figures have the same shape and different sizes.
  4. D)D. Similar figures have different shapes and different sizes, while congruent figures have the same shape and same size.

Step-by-step solution

  1. Congruent figures are identical in both shape and size. They can be perfectly superimposed on each other.
  2. Similar figures have the same shape but can differ in size. Their corresponding angles are equal, and their corresponding sides are in proportion.
  3. Therefore, the statement that similar figures have the same shape but not necessarily the same size, and congruent figures have both the same shape and the same size, is correct.

Answer: B. Similar figures have the same shape but not necessarily the same size, while congruent figures have both the same shape and the same size.

Example 2medium

Which of the following statements is TRUE regarding two similar figures?
  1. A)They must have the same size.
  2. B)They must have the same shape.
  3. C)They must have the same area.
  4. D)They must be congruent.

Step-by-step solution

  1. Similar figures are figures that have the same shape but not necessarily the same size.
  2. Congruent figures have both the same shape and the same size.
  3. Therefore, the only true statement is that similar figures must have the same shape.

Answer: They must have the same shape.

Example 3hard

In a trapezium ABCD, AB is parallel to DC. Its diagonals AC and BD intersect at O. If AB = 3CD and the area of ΔAOB is 72 cm², what is the area of ΔCOD?
  1. A)8 cm²
  2. B)12 cm²
  3. C)18 cm²
  4. D)24 cm²

Step-by-step solution

  1. Since AB || DC, by AA similarity criterion, ΔAOB ~ ΔCOD (∠AOB = ∠COD (vertically opposite angles), ∠OAB = ∠OCD (alternate interior angles)).
  2. The ratio of areas of similar triangles is the square of the ratio of their corresponding sides. So, Area(ΔAOB) / Area(ΔCOD) = (AB/CD)².
  3. Given AB = 3CD, so AB/CD = 3/1. Therefore, Area(ΔAOB) / Area(ΔCOD) = (3/1)² = 9/1.
  4. Given Area(ΔAOB) = 72 cm², we have 72 / Area(ΔCOD) = 9. Solving for Area(ΔCOD), we get Area(ΔCOD) = 72 / 9 = 8 cm².

Answer: 8 cm²

Practice questions on Similarity

  1. Q1.easy

    In two triangles, ΔABC and ΔPQR, if ∠A = 60°, ∠B = 80°, ∠P = 60°, and ∠Q = 80°, which similarity criterion can be used to conclude that ΔABC ~ ΔPQR?
    1. A)A. SSS Similarity
    2. B)B. SAS Similarity
    3. C)C. AA Similarity
    4. D)D. BPT
    Show answer

    Answer: C. AA Similarity

    Hint: Consider the information given about the angles in both triangles. If two pairs of corresponding angles are equal, what does that imply?

  2. Q2.easy

    Consider two triangles, ΔXYZ and ΔMNO. If XY/MN = YZ/NO = XZ/MO = 3/2, which similarity criterion applies?
    1. A)A. AA Similarity
    2. B)B. SAS Similarity
    3. C)C. SSS Similarity
    4. D)D. RHS Similarity
    Show answer

    Answer: C. SSS Similarity

    Hint: Look at the given information: all three pairs of corresponding sides are in the same ratio. Which criterion relies on this?

  3. Q3.easy

    In ΔABC, a line DE is drawn parallel to BC, intersecting AB at D and AC at E. If AD = 4 cm, DB = 6 cm, and AE = 3 cm, what is the length of EC?
    1. A)A. 4.5 cm
    2. B)B. 2 cm
    3. C)C. 3 cm
    4. D)D. 5 cm
    Show answer

    Answer: A. 4.5 cm

    Hint: Recall the Basic Proportionality Theorem (BPT), also known as Thales Theorem. It states how parallel lines divide the sides of a triangle proportionally.

  4. Q4.medium

    In ΔABC and ΔPQR, if ∠A = 40°, ∠B = 80°, ∠P = 40° and ∠R = 70°, then which of the following is true?
    1. A)ΔABC ~ ΔPQR
    2. B)ΔABC ~ ΔQPR
    3. C)ΔABC ~ ΔRPQ
    4. D)ΔABC is not similar to ΔPQR
    Show answer

    Answer: ΔABC is not similar to ΔPQR

    Hint: First, calculate the missing angles in both triangles. Then, try to establish a one-to-one correspondence between vertices such that all corresponding angles are equal.

  5. Q5.medium

    In ΔABC, D and E are points on AB and AC respectively such that DE || BC. If AD = 4 cm, DB = 6 cm and AE = 5 cm, what is the length of EC?
    1. A)7.5 cm
    2. B)6 cm
    3. C)8 cm
    4. D)10 cm
    Show answer

    Answer: 7.5 cm

    Hint: Recall the Basic Proportionality Theorem (BPT), also known as Thales Theorem, which applies when a line is parallel to one side of a triangle.

  6. Q6.medium

    In ΔPQR, points S and T are on sides PQ and PR respectively. Which of the following conditions ensures ST || QR?
    1. A)PS/PQ = PT/PR
    2. B)PS/SQ = PT/TR
    3. C)PS/SQ = PQ/PR
    4. D)PT/TR = PQ/PS
    Show answer

    Answer: PS/SQ = PT/TR

    Hint: Think about the Converse of the Basic Proportionality Theorem. It specifies the ratio that must hold for the line segment to be parallel.

  7. Q7.hard

    In ΔABC, D and E are points on AB and AC respectively such that DE || BC. If AD:DB = 2:3 and F is a point on DE such that AF is a median of ΔADE, and the area of ΔABC is 125 cm², what is the area of ΔADF?
    1. A)10 cm²
    2. B)12 cm²
    3. C)15 cm²
    4. D)20 cm²
    Show answer

    Answer: 10 cm²

    Hint: First, find the ratio AD/AB. Then, use the property of areas of similar triangles. Remember that a median divides a triangle into two triangles of equal area.

  8. Q8.hard

    In a right-angled triangle ABC, with ∠B = 90°, BD is perpendicular to AC. Which of the following relationships is always true?
    1. A)BD² = AD × DC
    2. B)AB² = AD × AC
    3. C)BC² = CD × AC
    4. D)All of the above
    Show answer

    Answer: All of the above

    Hint: Consider the three triangles formed (ΔADB, ΔBDC, ΔABC). Prove their similarity to each other and deduce the proportional sides.

  9. Q9.hard

    A lamp post AB is 6 m high. A boy CD, 1.4 m tall, stands 8 m away from the base of the lamp post. A mirror is placed on the ground between the boy and the lamp post at point M, such that the boy can see the top of the lamp post in the mirror. If the boy's eyes are at point C, which is 1.4 m above the ground, and by the laws of reflection, the angle of incidence equals the angle of reflection (∠AMB = ∠CMD is NOT correct. Angle of incidence = Angle of reflection means ∠AMB and ∠CMD are NOT equal; rather, if the light ray from B hits M and reflects to C, then ∠BMA = ∠CMD is NOT correct. It means angle between BM and normal = angle between CM and normal. Which means the angle BMA and CMA are equal to the angle between the vertical line at M and the rays). This means ΔABM ~ ΔCDM. What is the distance of the mirror from the base of the lamp post (AM)?
    1. A)5.5 m
    2. B)6 m
    3. C)6.4 m
    4. D)7 m
    Show answer

    Answer: 6 m

    Hint: The angles of incidence and reflection being equal implies that the angle formed by the light ray from the top of the lamp post to the mirror is equal to the angle formed by the reflected ray from the mirror to the boy's eye, both with respect to the ground. This forms two similar right-angled triangles.

These are 9 of the 47 questions available for Similarity. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.