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Triangles

NCERT Class 10 · Mathematics Based on NCERT Class 10 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter builds on the concept of congruence from Class IX to introduce similarity, focusing on figures that have the same shape but not necessarily the same size. The central idea is that two triangles are similar if their corresponding angles are equal and their corresponding sides are in the same ratio, which leads to powerful theorems like the Basic Proportionality Theorem (Thales Theorem) and criteria for similarity (AAA, AA, SSS, SAS). The ultimate goal is to apply these principles to prove the Pythagoras Theorem and solve real-world problems involving indirect measurement, such as finding heights and distances.
Key Concepts (12)
Similar Figures
Two figures having the same shape but not necessarily the same size are called similar figures.
e.g. All circles are similar because they have the same shape, even if their radii are different.
Congruent Figures
Two figures are said to be congruent if they have the same shape and the same size.
e.g. Two squares with the same side length are congruent.
Similar Polygons
Two polygons of the same number of sides are similar if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio (or proportion).
e.g. Two quadrilaterals ABCD and A'B'C'D' are similar if ∠A = ∠A', ∠B = ∠B', ∠C = ∠C', ∠D = ∠D' and AB/A'B' = BC/B'C' = CD/C'D' = DA/D'A'.
Scale Factor (or Representative Fraction)
The same ratio of the corresponding sides of similar polygons is referred to as the scale factor.
e.g. If a photograph is enlarged from 35mm to 45mm, the scale factor is 45/35.
Equiangular Triangles
If corresponding angles of two triangles are equal, then they are known as equiangular triangles.
e.g. Triangle ABC and triangle DEF are equiangular if ∠A = ∠D, ∠B = ∠E, and ∠C = ∠F.
Basic Proportionality Theorem (Thales Theorem)
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
e.g. In triangle ABC, if DE || BC and D is on AB, E is on AC, then AD/DB = AE/EC.
Converse of Basic Proportionality Theorem
If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
e.g. In triangle ABC, if AD/DB = AE/EC, then DE || BC.
AAA (Angle-Angle-Angle) Similarity Criterion
If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar.
e.g. If ∠A = ∠D, ∠B = ∠E, and ∠C = ∠F, then ΔABC ~ ΔDEF.
AA (Angle-Angle) Similarity Criterion
If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.
e.g. If ∠A = ∠D and ∠B = ∠E, then ΔABC ~ ΔDEF (the third angles will automatically be equal).
SSS (Side-Side-Side) Similarity Criterion
If in two triangles, sides of one triangle are proportional to (i.e., in the same ratio of) the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar.
e.g. If AB/DE = BC/EF = CA/FD, then ΔABC ~ ΔDEF.
SAS (Side-Angle-Side) Similarity Criterion
If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar.
e.g. If ∠A = ∠D and AB/DE = AC/DF, then ΔABC ~ ΔDEF.
RHS Similarity Criterion
If in two right triangles, the hypotenuse and one side of one triangle are proportional to the hypotenuse and one side of the other triangle, then the two triangles are similar.
e.g. In right triangles ABC and PQR, if ∠B = ∠Q = 90° and AC/PR = AB/PQ, then ΔABC ~ ΔPQR.
Formulas & Equations (4)
AD/DB = AE/EC
Reads as: AD divided by DB equals AE divided by EC
Variables: AD = length of segment from vertex A to point D on side AB; DB = length of segment from D to B; AE = length of segment from A to point E on side AC; EC = length of segment from E to C.
Use when: Applying the Basic Proportionality Theorem when a line parallel to BC intersects sides AB and AC at D and E respectively.
⚠ Students often incorrectly write AD/AB = AE/AC directly without first deriving it from AD/DB = AE/EC.
AB/DE = BC/EF = CA/FD
Reads as: AB divided by DE equals BC divided by EF equals CA divided by FD
Variables: AB, BC, CA are sides of triangle ABC; DE, EF, FD are corresponding sides of triangle DEF.
Use when: Checking SSS similarity criterion or stating the proportionality of corresponding sides in similar triangles.
⚠ Students forget to maintain the correct correspondence of vertices and write AB/EF or BC/FD, which is incorrect.
AB/DE = AC/DF
Reads as: AB divided by DE equals AC divided by DF
Variables: AB, AC are sides including angle A in triangle ABC; DE, DF are sides including angle D in triangle DEF.
Use when: Applying the SAS similarity criterion when one angle is equal and the sides including it are proportional.
⚠ Students apply this criterion when the equal angle is not the included angle between the proportional sides.
CA² = CB × CD
Reads as: CA squared equals CB multiplied by CD
Variables: CA = side of triangle ABC; CB = side of triangle ABC; CD = segment on side BC from C to point D.
Use when: Proving a relationship when D is a point on BC such that ∠ADC = ∠BAC (as in Exercise 6.3, Q13).
⚠ Students forget to square CA and write CA = CB × CD instead.
How It Works
Proving Basic Proportionality Theorem (Thales Theorem)
  1. Step 1: Draw triangle ABC with line DE parallel to BC intersecting AB at D and AC at E.
  2. Step 2: Join BE and CD. Draw DM perpendicular to AC and EN perpendicular to AB.
  3. Step 3: Express areas: ar(ADE) = 1/2 × AD × EN, ar(BDE) = 1/2 × DB × EN, ar(ADE) = 1/2 × AE × DM, ar(DEC) = 1/2 × EC × DM.
  4. Step 4: Form ratios: ar(ADE)/ar(BDE) = AD/DB and ar(ADE)/ar(DEC) = AE/EC.
  5. Step 5: Note that triangles BDE and DEC are on the same base DE and between the same parallels BC and DE, so ar(BDE) = ar(DEC).
  6. Step 6: Equate the ratios from steps 4 and 5 to get AD/DB = AE/EC.
💡 The proof uses areas of triangles, not side lengths directly. Students often forget to state that triangles BDE and DEC have equal areas because they share the same base and lie between the same parallels.
Proving Converse of Basic Proportionality Theorem
  1. Step 1: Assume a line DE such that AD/DB = AE/EC, but DE is not parallel to BC.
  2. Step 2: Draw a line DE' parallel to BC through D intersecting AC at E'.
  3. Step 3: By the Basic Proportionality Theorem, AD/DB = AE'/E'C.
  4. Step 4: Given AD/DB = AE/EC, we get AE/EC = AE'/E'C.
  5. Step 5: Add 1 to both sides: (AE/EC) + 1 = (AE'/E'C) + 1, giving AC/EC = AC/E'C, so EC = E'C.
  6. Step 6: Since E and E' lie on the same line AC and EC = E'C, E and E' must coincide, proving DE is parallel to BC.
💡 The proof uses a contradiction approach. Students often skip the step of adding 1 to both sides to show that E and E' coincide.
Applying Similarity to Find Shadow Length (Example 7)
  1. Step 1: Identify the lamp-post (AB) and the girl (CD) as vertical objects, so ∠B = ∠D = 90°.
  2. Step 2: Note that the girl walks away from the lamp-post at 1.2 m/s for 4 seconds, so BD = 1.2 × 4 = 4.8 m.
  3. Step 3: Let the shadow length DE = x metres.
  4. Step 4: Observe that triangles ABE and CDE share angle E, and both have a right angle, so ΔABE ~ ΔCDE by AA similarity.
  5. Step 5: Write the proportion: BE/DE = AB/CD, i.e., (4.8 + x)/x = 3.6/0.9.
  6. Step 6: Solve: (4.8 + x)/x = 4, so 4.8 + x = 4x, 3x = 4.8, x = 1.6 m.
💡 Students often forget to convert units (90 cm to 0.9 m) and incorrectly set up the proportion, e.g., writing AB/CD = DE/BE instead of the correct BE/DE = AB/CD.
Exam Traps — Don't Make These Mistakes
⚠ Confusing Similarity and Congruence
✗ All similar figures are congruent.
→ Similar figures have the same shape but not necessarily the same size, while congruent figures have both the same shape and size. Congruence is a special case of similarity, not the other way around.
✓ All congruent figures are similar, but similar figures need not be congruent.
⚠ Misapplying the Basic Proportionality Theorem
✗ In triangle ABC with DE || BC, AD/AB = AE/AC directly without proof.
→ The theorem states AD/DB = AE/EC. The relation AD/AB = AE/AC is a derived result that must be shown by adding 1 to both sides of the original proportion.
✓ Since DE || BC, by BPT, AD/DB = AE/EC. Then, DB/AD = EC/AE. Adding 1 to both sides gives (DB+AD)/AD = (EC+AE)/AE, so AB/AD = AC/AE, hence AD/AB = AE/AC.
⚠ Incorrect Correspondence in Similar Triangles
✗ ΔABC ~ ΔDEF implies AB/EF = BC/DE.
→ The correspondence of vertices must be maintained. If ΔABC ~ ΔDEF, then A↔D, B↔E, C↔F, so AB/DE = BC/EF = CA/FD.
✓ ΔABC ~ ΔDEF implies AB/DE = BC/EF = CA/FD.
⚠ Using SAS Similarity with Non-Included Angle
✗ If ∠A = ∠D and AB/DE = BC/EF, then ΔABC ~ ΔDEF.
→ SAS similarity requires that the proportional sides include the equal angle. Here, AB and BC include angle B, not angle A. The correct pair for angle A is sides AB and AC.
✓ If ∠A = ∠D and AB/DE = AC/DF, then ΔABC ~ ΔDEF by SAS similarity criterion.
⚠ Forgetting to State 'Corresponding' in Definitions
✗ Two polygons are similar if their angles are equal and sides are proportional.
→ The definition requires that corresponding angles are equal and corresponding sides are in the same ratio. Without the word 'corresponding', the statement is incomplete and can lead to incorrect conclusions.
✓ Two polygons of the same number of sides are similar if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio (or proportion).
Compare & Contrast
Number of angle pairs required
AAA Similarity Criterion
AA Similarity Criterion
AAA requires all three pairs of corresponding angles to be equal, while AA requires only two pairs because the third pair will automatically be equal by the angle sum property of a triangle.
Condition on sides
SSS Similarity Criterion
SSS Congruence Criterion
SSS similarity requires that the corresponding sides are in the same ratio (proportional), not necessarily equal. SSS congruence requires that the corresponding sides are exactly equal.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 Short Answer (3 marks), 1 Long Answer (5 marks) — total 10 marks
Most asked: Basic Proportionality Theorem and its converse · AAA/AA similarity criterion · SSS similarity criterion · SAS similarity criterion · Problems involving indirect measurement (heights and shadows) · Proofs involving similarity of triangles
Don't skip: Proof-based questions on Basic Proportionality Theorem or its converse; Application problems involving similarity to find unknown lengths (e.g., shadow problems, pole and tower problems); Proving two triangles are similar using a specific criterion and then finding a side or angle
One-Line Revision (read aloud before the exam)
  1. Two figures having the same shape but not necessarily the same size are called similar figures.
  2. All congruent figures are similar, but similar figures need not be congruent.
  3. Two polygons of the same number of sides are similar if their corresponding angles are equal and their corresponding sides are in the same ratio.
  4. If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio (Basic Proportionality Theorem).
  5. If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side (Converse of BPT).
  6. AAA similarity criterion: If corresponding angles are equal, then triangles are similar.
  7. AA similarity criterion: If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar.
  8. SSS similarity criterion: If corresponding sides are in the same ratio, then triangles are similar.
  9. SAS similarity criterion: If one angle is equal and the sides including it are proportional, then triangles are similar.
  10. RHS similarity criterion: In right triangles, if the hypotenuse and one side are proportional, the triangles are similar.
  11. The ratio of any two corresponding sides in two equiangular triangles is always the same.
  12. In similar triangles, corresponding angles are equal and corresponding sides are proportional.
  13. The scale factor is the common ratio of corresponding sides of similar polygons.
  14. A line through the mid-point of one side of a triangle parallel to another side bisects the third side.
  15. The line joining the mid-points of any two sides of a triangle is parallel to the third side.

MCQs — 10 Questions with Answers

Q1. A photographer enlarges a 35mm photograph to 70mm size. If a line segment in the original photograph measures 5cm, what can be said about the corresponding line segment in the enlarged photograph using the concept of similar figures?

  • A. It measures 10cm because all corresponding linear dimensions scale by the same factor (ratio 35:70 = 1:2) ✓
  • B. It measures 10cm, but only if all angles in both photographs are also equal
  • C. It cannot be determined without knowing the angles of the figure
  • D. It measures 5cm because similarity preserves actual side lengths

Answer: A — Similar figures have corresponding sides in the same ratio; the scale factor 35:70 = 1:2 means all linear dimensions double, independent of angles being equal (which are automatically equal in photographs of the same object).

Q2. Two quadrilaterals have all four corresponding angles equal. A student claims they must be similar. Is this claim justified based on the definition of similarity of polygons given in the chapter?

  • A. Yes, because equal corresponding angles guarantee similarity for all polygons
  • B. No, because similarity requires BOTH equal corresponding angles AND corresponding sides in the same ratio ✓
  • C. Yes, but only if the quadrilaterals are convex
  • D. No, because angles alone can never determine similarity

Answer: B — The chapter explicitly states similarity requires two conditions: (i) equal corresponding angles AND (ii) corresponding sides in the same ratio; a square and rectangle have equal corresponding angles but unequal side ratios, so they are not similar.

Q3. Assertion (A): All circles are similar to each other. Reason (R): All circles have the same shape but not necessarily the same size. Choose the correct option:

  • A. Both A and R are true and R is the correct explanation of A ✓
  • B. Both A and R are true but R is not the correct explanation of A
  • C. A is true but R is false
  • D. A is false but R is true

Answer: A — Both statements are true; Reason R explains Assertion A because the definition of similar figures explicitly requires same shape (not necessarily same size), which is precisely why all circles of different radii are similar.

Q4. A student observes that all congruent figures are similar. Based on the chapter's definitions, can the converse also be true—that all similar figures are congruent?

  • A. Yes, because congruence and similarity are equivalent concepts
  • B. No, because similar figures have the same shape but not necessarily the same size, while congruent figures must have both same shape and same size ✓
  • C. Yes, but only for triangles
  • D. No, because congruence is only defined for angles, not sides

Answer: B — The chapter explicitly states 'all congruent figures are similar but similar figures need not be congruent'; congruence requires identical size and shape, while similarity only requires the same shape.

Q5. Assertion (A): A rectangle and a square cannot be similar. Reason (R): They have different numbers of sides. Choose the correct option:

  • A. Both A and R are true and R is the correct explanation of A
  • B. Both A and R are true but R is not the correct explanation of A
  • C. A is true but R is false ✓
  • D. A is false but R is true

Answer: C — Assertion A is true (rectangles and squares are not similar because side ratios differ), but Reason R is false (both have 4 sides); the correct reason is that corresponding sides are not in the same ratio, as shown in Fig. 6.6.

Q6. In Activity 1 of the chapter, a cardboard quadrilateral ABCD is placed between a bulb and a table to cast a shadow ABCD on the table. What property of light and geometry ensures that the shadow is similar to the original quadrilateral?

  • A. Light travels in straight lines, causing vertices A, B, C, D to lie on rays OA, OB, OC, OD respectively, preserving angle measures and creating proportional side lengths ✓
  • B. The shadow is similar because light always creates congruent images regardless of distance
  • C. Similarity is guaranteed only if the cardboard is positioned at exactly 45° to the ground
  • D. The angles are preserved but side lengths are not related unless we measure all sides

Answer: A — Light's straight-line propagation causes the shadow to be a magnified (or scaled) version of the original with corresponding vertices collinear with the light source, preserving both angles and side ratios—the exact conditions for similarity.

Q7. Assertion (A): If the scale factor between two similar polygons is 2:1, then each side of the first polygon is twice the corresponding side of the second polygon. Reason (R): The scale factor represents the ratio of corresponding sides in similar figures. Choose the correct option:

  • A. Both A and R are true and R is the correct explanation of A ✓
  • B. Both A and R are true but R is not the correct explanation of A
  • C. A is true but R is false
  • D. A is false but R is true

Answer: A — Both A and R are true; the scale factor 2:1 directly means corresponding sides have ratio 2:1, making sides of the first polygon twice those of the second—R correctly explains why A is true.

Q8. Two figures have the same shape but one is a 3D solid and the other is a 2D drawing. Can these figures be similar according to the definition given in the chapter?

  • A. Yes, because both have the same shape
  • B. No, because the definition of similar figures applies to polygons and figures of the same dimension; a 3D solid and 2D drawing belong to different geometric categories ✓
  • C. Yes, if all angles are equal
  • D. No, because 3D objects cannot be similar to anything

Answer: B — The chapter defines similarity for polygons (2D figures) based on corresponding angles and sides; comparing a 3D solid to a 2D figure violates this definition because they are not comparable geometric objects.

Q9. A student claims: 'If polygon P is similar to polygon Q with scale factor 3:1, and polygon Q is similar to polygon R with scale factor 2:1, then polygon P is similar to polygon R with scale factor 6:1.' Is this reasoning correct?

  • A. Yes, the scale factors multiply correctly: 3:1 × 2:1 = 6:1, and the remark in the chapter confirms transitivity of similarity ✓
  • B. No, because scale factors cannot be multiplied; we must measure polygon R directly
  • C. Yes, but only if P, Q, and R are all triangles
  • D. No, because similarity is not a transitive relation

Answer: A — The chapter's remark explicitly states that if one polygon is similar to another and that polygon is similar to a third, then the first is similar to the third; composing scale factors 3:1 and 2:1 gives 6:1.

Q10. Assertion (A): Two quadrilaterals with all corresponding sides in the ratio 1:1 must be congruent. Reason (R): When corresponding sides are in the ratio 1:1, all sides of both quadrilaterals are equal in length. Choose the correct option:

  • A. Both A and R are true and R is the correct explanation of A
  • B. Both A and R are true but R is not the correct explanation of A ✓
  • C. A is true but R is false
  • D. A is false but R is true

Answer: B — Both A and R are true (equal ratios mean equal side lengths), but this alone does not guarantee congruence unless angles are also equal; R describes a necessary but not sufficient condition for the conclusion in A.

Flashcards

What is the definition of similar figures?

Two figures are similar if they have the same shape but not necessarily the same size.

State the two conditions for two polygons to be similar.

Their corresponding angles must be equal AND their corresponding sides must be in the same ratio (proportion).

Are all congruent figures similar? Explain.

Yes, all congruent figures are similar because they have both the same shape and same size, which satisfies the similarity conditions.

Are all similar figures congruent?

No, similar figures need not be congruent because they can have different sizes.

Define scale factor (Representative Fraction).

The scale factor is the constant ratio between corresponding sides of two similar polygons.

Can a square and a rectangle be similar? Why or why not?

No, because although their corresponding angles are equal (all 90°), their corresponding sides are not in the same ratio.

Can a square and a rhombus be similar? Why or why not?

No, because although their corresponding sides are in the same ratio, their corresponding angles are not equal.

Are all circles similar to each other?

Yes, all circles are similar because they have the same shape regardless of their radius.

Are all equilateral triangles similar to each other?

Yes, all equilateral triangles are similar because they have all angles equal (60° each) and sides in the same ratio.

What real-life application uses the concept of similar figures?

Photographers enlarge or reduce photographs using a fixed scale factor to maintain the same shape at different sizes.

Important Board Questions

Define similar figures. Give one example of a pair of similar figures and one example of a pair of non-similar figures. [2 marks]

State that similar figures have same shape but not necessarily same size. For examples: similar pair could be two circles of different radii OR two equilateral triangles of different side lengths; non-similar pair could be a square and rectangle OR a circle and triangle.

Two quadrilaterals ABCD and PQRS are similar with a scale factor of 1:3. If AB = 5 cm, BC = 7 cm, and CD = 6 cm, find the corresponding sides PQ, QR, and RS of quadrilateral PQRS. Justify your answer. [3 marks]

Use the definition that corresponding sides of similar polygons are in the same ratio (scale factor). If scale factor is 1:3, then each side of PQRS = 3 × corresponding side of ABCD. Apply this formula to each side: PQ = 3 × AB = 15 cm, QR = 3 × BC = 21 cm, RS = 3 × CD = 18 cm.

Explain with reasons why a square and a rectangle are NOT similar figures, even though they appear to have related properties. Use a specific example with measurements to support your explanation. [5 marks]

State that similarity requires BOTH conditions: (1) corresponding angles equal AND (2) corresponding sides in same ratio. Show that while a square and rectangle both have all 90° angles (condition 1 satisfied), their sides are NOT in the same ratio (condition 2 fails). Use example: square with all sides 4 cm vs rectangle with sides 4 cm and 8 cm—angles are equal but side ratios are 4:4 vs 4:8, which are different. Conclude that having only one condition satisfied is insufficient for similarity.

Assertion-Reason Questions for Triangles

Q1. Assertion (A):

All equilateral triangles are similar to each other.

Reason (R):

All equilateral triangles have equal angles of 60° each and their sides are always in the same ratio to each other.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. The reason correctly explains why all equilateral triangles are similar - they all have identical angles (60°-60°-60°) and maintain the same proportional relationship between sides. This is a complete and valid explanation for the assertion.

Q2. Assertion (A):

Two figures can be similar even if one is a circle and the other is a square.

Reason (R):

Similarity is defined only by having the same shape, regardless of the number of sides or curves.

Answer: D — A is false but R is true — Both the assertion and reason are false. A circle and a square have completely different shapes and cannot be similar. Similarity requires not only the same general appearance but also that figures must be of the same geometric category - you cannot compare a curve-based figure with a polygon-based figure for similarity.

Q3. Assertion (A):

If quadrilateral ABCD is similar to quadrilateral PQRS with a scale factor of 2:1, then the area of ABCD is twice the area of PQRS.

Reason (R):

Linear scale factor and area scale factor are always equal.

Answer: C — A is true but R is false — The assertion is true - with linear scale factor 2:1, the area ratio becomes (2)² = 4:1, so ABCD has 4 times the area of PQRS, not 2 times. However, the reason is false because the area scale factor is the SQUARE of the linear scale factor, not equal to it. The assertion happens to be false due to incorrect reasoning.

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Case-Based Questions (CBQ) for Triangles

Case Study:

A photographer has taken a photograph on a 35mm film and wants to create prints of different sizes. When enlarging to 45mm, every line segment in the original is enlarged by a factor of 45/35. She also plans to create a postcard-sized print at 55mm. A landscape feature that measures 3.5 cm in the 35mm original needs to be measured in the 45mm enlargement. The photographer wants to ensure all enlargements maintain the same proportions and angles as the original.

Q1. What is the scale factor when enlarging from 35mm to 45mm?

  • A. 35:45
  • B. 45:35 ✓
  • C. 9:7
  • D. 7:9

Answer: B

Q2. If a line segment measures 3.5 cm in the 35mm original, what will be its length in the 45mm enlargement?

  • A. 2.72 cm
  • B. 4.5 cm ✓
  • C. 4.58 cm
  • D. 7 cm

Answer: B

Q3. Which property ensures that the enlarged photograph is similar to the original?

  • A. All sides are increased by the same factor and all angles remain equal ✓
  • B. The perimeter is enlarged proportionally
  • C. The area becomes larger
  • D. The print size matches the original

Answer: A

Q4. If the 55mm enlargement scale factor from 35mm is 55:35, how many times is this print enlarged?

  • A. 1.57 times ✓
  • B. 1.29 times
  • C. 1.75 times
  • D. 2 times

Answer: A

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Previous Year Questions (PYQ) for Triangles

Real CBSE board exam questions from past papers — free to attempt.

CBSE 2026 · Set 31/4/1 · 1 mark

In the given figure, PQ || BC. If AP : AB = 3 : 7 then, AQ : QC equals:

  • 3 : 7
  • 3 : 10
  • 7 : 3
  • 3 : 4

CBSE 2026 · Set 31/4/1 · 2 marks

In the given figure, QR || CB and RP || AC. If BR = 10 cm, QA = 12 cm, BP = 12 cm and PC = 18 cm, then find the lengths of AR and QC.

CBSE 2026 · Set 31/5/1 · 1 mark

Which of the following statements is not always true?

  • Two circles are similar
  • Two isosceles right triangles are similar
  • Two rectangles are similar
  • Two equilateral triangles are similar

CBSE 2026 · Set 31/5/1 · 1 mark

In the given figure, DE || BC. If AD : AB = 1 : 3 and AE = 2.5 cm, then AC equals

  • 7.5 cm
  • 5 cm
  • 10 cm
  • 2.5 cm

CBSE 2026 · Set 31/5/1 · 2 marks

In the given figure, AB || DC. If OB = 3OD and CD = 1.8 cm, then find the length AB.

CBSE 2026 · Set 31/5/1 · 5 marks

In the given figure, ABC is right angled triangle with ∠A = 90°. AD is perpendicular to BC. Prove that: (i) △DBA ~ △DAC (ii) DA² = DB × DC (iii) Find the area of △ABC when DB = 9 cm and DC = 16 cm.

CBSE 2026 · Set 31/5/1 · 1 mark

In a circular museum hall of radius 14 m, some statues are displayed. Statues are kept inside the inner concentric circle of radius 7 m. One such statue lying in sector OAB, is fenced along line segments OA, AP, PB and BO where P is a point on outer circle.

Prove that △OAP ≅ △OBP.

CBSE 2026 · Set 31/5/2 · 2 marks

In the given figure, △ODC ~ △OBA. If ∠BOC = 110º, ∠ODC = 45º and AB = 2 CD, then find (i) m∠OAB (ii) OB : OD.

CBSE 2026 · Set 31/5/2 · 5 marks

(a) In the given figure, △ABC is right angled triangle with ∠A = 90º. AD is perpendicular to BC. Prove that: (i) △DBA ~ △DAC (ii) DA² = DB × DC (iii) Find the area of △ABC when DB = 9 cm and DC = 16 cm. OR (b) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio.

CBSE 2026 · Set 31/5/3 · 2 marks

In the given figure, two triangles ABC and PQR are shown such that ∠A = ∠P and ∠C = ∠R. If AD ⊥ BC and PS ⊥ QR, then prove that (i) △ADB ~ △PSQ (ii) AD × QS = BD × PS.

CBSE 2026 · Set 31/5/3 · 5 marks

In the given figure, ABC is right angled triangle with ∠A = 90°. AD is perpendicular to BC. Prove that: (i) △DBA ~ △DAC (ii) DA² = DB × DC (iii) Find the area of △ABC when DB = 9 cm and DC = 16 cm.

CBSE 2026 · Set 31/1164/30 · 1 mark

ABCD is a trapezium in which AB||DC and E, F are points on AD and BC respectively such that EF||DC. If ED = 36 cm, BF = 70 cm and FC = 30 cm, then the length of AD is :

  • 124 cm
  • 120 cm
  • 110 cm
  • 114 cm

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Class 10 Maths Formula Sheet

Every formula from all 14 chapters on one page, checked against the NCERT textbook and the CBSE 2026-27 syllabus. Free PDF download.

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Class 10 Mathematics Previous Year Questions, Chapter-wise

All 1107 CBSE board questions from 2022–2026, sorted by chapter and question type. Free PDF download.

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