Geometry
Triangles
Similarity, BPT, Pythagoras — and how examiners ask about them.
9 marks
Board exam weightage (out of 80)
108
Worked examples on this page
Geometry
Unit
About this chapter
Triangles is the highest-weightage Geometry chapter on the CBSE Class 10 board exam, typically worth around 15 marks across the Geometry unit alongside Circles. The chapter covers similarity of triangles, the Basic Proportionality Theorem (BPT) and its converse, similarity criteria (AA, SAS, SSS), Pythagoras' theorem and its converse, and the ratio of areas of similar triangles. Most board questions ask you to prove a similarity result and use it to compute a length, or to apply BPT to find an unknown segment in a parallel-lines configuration.
Key topics covered
- •Similar triangles and the AA, SAS, SSS criteria
- •Basic Proportionality Theorem (BPT) and its converse
- •Pythagoras' theorem and its converse
- •Ratio of areas of similar triangles
- •Medians, altitudes, and similar-triangles configurations
Common question types in board exam
The patterns that recur most often across recent CBSE papers in this chapter.
similar triangles· 140 questionsbasic proportionality theorem· 73 questionsparallel lines· 61 questionssimilarity· 25 questionsproportionality· 23 questionsratio· 22 questions
Question difficulty mix
Count of questions seen across all CBSE papers we've indexed, by mark value.
116×1-mark
55×2-mark
3×3-mark
7×4-mark
61×5-mark
Worked solutions (108)
Grouped by year and tier2026 standard
- In ABC, DE BC. If AD = , DB = , AE = and EC = , then find the…Set 1 Part 1 · Q22 · 2 marks
- State and prove Basic Proportionality Theorem.Set 1 Part 1 · Q34 · 5 marks
- In , . If , , and , then find the va…Set 1 Part 2 · Q21 · 2 marks
- State and prove Basic Proportionality Theorem.Set 1 Part 2 · Q33 · 5 marks
- In ABC, DE BC. If AD = , DB = , AE = and EC = , then find the…Set 1 Part 3 · Q25 · 2 marks
- State and prove Basic Proportionality Theorem.Set 1 Part 3 · Q32 · 5 marks
- (a) In the given figure, . If AK = 10 cm, BC = 3.5 cm and HK = 7 cm, find the leng…Set 2 Part 1 · Q22 · 2 marks
- (a) Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides in disti…Set 2 Part 1 · Q34 · 5 marks
- (a) In the given figure, . If cm, cm and cm, f…Set 2 Part 2 · Q21 · 2 marks
- (a) Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides in disti…Set 2 Part 3 · Q32 · 5 marks
- The diagonals of a quadrilateral ABCD intersect each other at the point O such that…Set 3 Part 1 · Q24 · 2 marks
- Two right triangles and are drawn on the same hypotenuse . If and intersect at , …Set 3 Part 2 · Q22 · 2 marks
- The following distribution shows the weekly pocket allowance (in ₹) of some children of a locality. The mean p…Set 3 Part 2 · Q34 · 5 marks
- D is a point on the side BC of such that . Show that…Set 4 Part 1 · Q22 · 2 marks
- Carom board is a very popular game. The board is a square of side length 65 cm. It has circular pockets in eac…Set 4 Part 1 · Q36 · 4 marks
- In the given figure, and . Prove that : .Set 4 Part 2 · Q22 · 2 marks
- Carom board is a very popular game. The board is a square of side length 65 cm. It has circular pockets in eac…Set 4 Part 2 · Q38 · 4 marks
- In the given figure, is a square. is right angle triangle with . Prove …Set 4 Part 3 · Q23 · 2 marks
- In the given figure, AB || DE and AC || DF. Show that . If BC = 10 cm, EB = …Set 5 Part 1 · Q23 · 2 marks
- (a) D is the mid-point of side BC of . CE and BF intersect at O, a point on AD. AD is produced …Set 5 Part 1 · Q33 · 5 marks
- In the given figure, and . Show that . If…Set 5 Part 2 · Q21 · 2 marks
- (a) D is the mid-point of side BC of . CE and BF intersect at O, a point on AD. AD is produced …Set 5 Part 2 · Q34 · 5 marks
- In the given figure, and . Show that . If…Set 5 Part 3 · Q24 · 2 marks
- (a) D is the mid-point of side BC of . CE and BF intersect at O, a point on AD. AD is produced …Set 5 Part 3 · Q32 · 5 marks
2026 basic
- In a quadrilateral , diagonals and intersect each other at such that…Set 1 Part 1 · Q33 · 5 marks
- In the given figure, , and .…Set 1 Part 2 · Q25 · 2 marks
- State "Basic Proportionality Theorem" and use it to prove the following: A line through the mid-point of one s…Set 1 Part 2 · Q32 · 5 marks
- In the given figure, , and .…Set 1 Part 3 · Q23 · 2 marks
- State the converse of "Basic Proportionality Theorem" and use it to prove the following:Set 1 Part 3 · Q34 · 5 marks
- Verify the relation between the zeroes and the coefficients of the quadratic polynomial .Set 2 Part 1 · Q22 · 2 marks
- State the SAS criteria of similarity of two triangles. In the given figure, it is given that OA · OC = OB · OD…Set 2 Part 1 · Q33 · 5 marks
- In the figure given below, and . Prove that…Set 2 Part 2 · Q25 · 2 marks
- State AA criterion of similarity of two triangles and use it to prove the following.Set 2 Part 2 · Q32 · 5 marks
- D is a point on side BC of such that . Prove that…Set 2 Part 3 · Q23 · 2 marks
- State and prove "Basic Proportionality Theorem."Set 2 Part 3 · Q34 · 5 marks
- Verify the relationship between the zeroes and the coefficients of the polynomial .Set 3 Part 1 · Q22 · 2 marks
- State and Prove "Basic Proportionality Theorem".Set 3 Part 1 · Q33 · 5 marks
- In the given figure, and . Prove that…Set 3 Part 2 · Q25 · 2 marks
- In the given figure, CM and RN are respectively, the medians of and . If…Set 3 Part 2 · Q32 · 5 marks
- In the given figure, CM and RN are respectively, the medians of and . If…Set 3 Part 3 · Q34 · 5 marks
- (a) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points,…Set 4 Part 1 · Q33 · 5 marks
- In the given figure, QR CB and RP AC. If BR = 10 cm, QA = 12 cm, BP = 12 cm and PC = 1…Set 4 Part 2 · Q23 · 2 marks
- AD and PS are respectively, the medians of and . If…Set 4 Part 2 · Q34 · 5 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 th…Set 4 Part 3 · Q23 · 2 marks
- (a) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points,…Set 4 Part 3 · Q35 · 5 marks
- In the given figure, . If and cm, then find the length AB.Set 5 Part 1 · Q25 · 2 marks
- (a) In the given figure, is right angled triangle with . AD is perpendicular t…Set 5 Part 1 · Q32 · 5 marks
- In the given figure, . If , and…Set 5 Part 2 · Q22 · 2 marks
- (a) In the given figure, is right angled triangle with . AD is perpendicular t…Set 5 Part 2 · Q34 · 5 marks
- In the given figure, two triangles ABC and PQR are shown such that and…Set 5 Part 3 · Q22 · 2 marks
- (a) In the given figure, is right angled triangle with . AD is perpendicu…Set 5 Part 3 · Q35 · 5 marks
2025 standard
- (a) If in which cm, cm, cm and cm, then…Set 1 Part 1 · Q24 · 2 marks
- (a) The diagonal BD of a parallelogram ABCD intersects the line segment AE at the point F, where E is any poin…Set 1 Part 1 · Q33 · 5 marks
- State "Basic Proportionality Theorem" and use it to prove the following:Set 1 Part 2 · Q32 · 5 marks
- (a) If in which cm, cm, cm and cm, then…Set 1 Part 3 · Q21 · 2 marks
- (a) The diagonal BD of a parallelogram ABCD intersects the line segment AE at the point F, where E is any poin…Set 1 Part 3 · Q35 · 5 marks
- Prove that a line drawn parallel to one side of a triangle to intersect the other two sides in distinct points…Set 2 Part 1 · Q34 · 5 marks
- In the given figure, O is the centre of a circle of radius 7 cm. AB is a chord of the circle. Find the perimet…Set 2 Part 2 · Q22 · 2 marks
- Prove that a line drawn parallel to one side of a triangle to intersect the other two sides in distinct points…Set 2 Part 2 · Q33 · 5 marks
- A class teacher has the following absentees record of students of a class.Set 2 Part 3 · Q32 · 5 marks
- The perimeter of an isosceles triangle is 32 cm. If each equal side is th of the base, find the a…Set 3 Part 1 · Q32 · 5 marks
- (a) In the given figure, PA, QB and RC are perpendicular to AC. If PA = x units, QB = y units and RC = z units…Set 3 Part 1 · Q34 · 5 marks
- A necklace is made up of wooden beads. Each bead is in the form of a sphere of diameter 4.2 mm. A cylinder is …Set 3 Part 3 · Q35 · 5 marks
- Find 'mean' and 'mode' of the following data:Set 4 Part 1 · Q33 · 5 marks
- Express the equation ; as a quadratic equati…Set 4 Part 2 · Q33 · 5 marks
- State basic proportionality theorem. Use it to prove the following: If three parallel lines l, m, n are inters…Set 4 Part 2 · Q34 · 5 marks
- The corresponding sides of and are in the ratio . and…Set 4 Part 3 · Q32 · 5 marks
- In a pre-primary class, a teacher put cards numbered to in a bowl. A student picked up a card at ran…Set 5 Part 1 · Q24 · 2 marks
- AD and PS are angle bisectors of and of triangles ABC and PQR. If…Set 5 Part 2 · Q21 · 2 marks
- State the converse of basic proportionality theorem.Set 5 Part 2 · Q34 · 5 marks
- In and , AD and PS are altitudes such that a…Set 5 Part 3 · Q23 · 2 marks
- State the basic proportionality theorem. Use the theorem to do the following:Set 5 Part 3 · Q32 · 5 marks
- In the adjoining figure, and , prove that…Set 6 Part 2 · Q21 · 2 marks
- In the adjoining figure, is a right triangle, right angled at A and . Prove that …Set 6 Part 3 · Q32 · 5 marks
2025 basic
- (a) In the given figure, if PQ RS, then prove that .Set 1 Part 1 · Q22 · 2 marks
- State "Basic Proportionality Theorem" and use it to prove the following:Set 1 Part 1 · Q33 · 5 marks
- (a) In the given figure, if PQ RS, then prove that .Set 1 Part 2 · Q25 · 2 marks
- State "Basic Proportionality Theorem" and use it to prove the following:Set 1 Part 2 · Q32 · 5 marks
- In the given figure, , and . Find the …Set 1 Part 3 · Q23 · 2 marks
- State the converse of "Basic Proportionality Theorem" and use it to prove the following:Set 1 Part 3 · Q34 · 5 marks
- D is a point on side BC of such that . Prove that…Set 2 Part 1 · Q22 · 2 marks
- State the SAS criteria of similarity of two triangles. In the given figure, it is given that…Set 2 Part 1 · Q33 · 5 marks
- D is a point on side BC of such that . Prove that…Set 2 Part 2 · Q25 · 2 marks
- State AA criterion of similarity of two triangles and use it to prove the following.Set 2 Part 2 · Q32 · 5 marks
- D is a point on side BC of such that . Prove that…Set 2 Part 3 · Q23 · 2 marks
- State and prove "Basic Proportionality Theorem."Set 2 Part 3 · Q34 · 5 marks
- (a) A vertical pole of height 10 m casts a shadow of 15 m on the ground and at the same time, a tower casts a …Set 3 Part 1 · Q22 · 2 marks
- (a) State and Prove "Basic Proportionality Theorem".Set 3 Part 1 · Q33 · 5 marks
- (a) A vertical pole of height 10 m casts a shadow of 15 m on the ground and at the same time, a tower casts a …Set 3 Part 2 · Q25 · 2 marks
- (a) State and Prove "Basic Proportionality Theorem".Set 3 Part 2 · Q32 · 5 marks
- In the given figure, CM and RN are respectively, the medians of and . If…Set 3 Part 3 · Q34 · 5 marks
- In the given figure and . Prove that …Set 4 Part 1 · Q25 · 2 marks
- In a , P and Q are points on AB and AC respectively such that . Prove that the…Set 4 Part 1 · Q34 · 5 marks
- In a , P and Q are points on AB and AC respectively such that . Prove that the…Set 4 Part 2 · Q32 · 5 marks
- The diagonal BD of parallelogram ABCD is divided by segment AE in the ratio 1 : 2. If cm, find the …Set 4 Part 3 · Q24 · 2 marks
- If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points then…Set 4 Part 3 · Q33 · 5 marks
- Point E lies on the extended side AD of parallelogram ABCD. BE intersects CD at F. Show that (i)…Set 5 Part 1 · Q31 · 3 marks
- (a) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points,…Set 5 Part 1 · Q33 · 5 marks
- Point E lies on the extended side AD of parallelogram ABCD. BE intersects CD at F. Show that (i)…Set 5 Part 2 · Q28 · 3 marks
- It is given that sides AB and AC and median AD of are respectively proportional to sides PQ an…Set 5 Part 2 · Q35 · 5 marks
- It is given that . Prove that .Set 5 Part 3 · Q29 · 3 marks
- It is given that sides AB and AC and median AD of are respectively proportional to sides PQ an…Set 5 Part 3 · Q35 · 5 marks
- In a Fine Arts class, students were asked to design triangular tiles in geometric pattern.Set 5 Part 3 · Q38 · 4 marks
- In a trapezium ABCD, and its diagonals intersect at O. Prove that…Set 6 Part 1 · Q22 · 2 marks
- A triangular window of a building is shown above. Its diagram represents a with…Set 6 Part 1 · Q36 · 4 marks
- A triangular window of a building is shown above. Its diagram represents a with…Set 6 Part 2 · Q37 · 4 marks
- The perimeters of two similar triangles are 22 cm and 33 cm respectively. If one side of first triangle is 9 c…Set 6 Part 3 · Q25 · 2 marks
- A triangular window of a building is shown above. Its diagram represents a with…Set 6 Part 3 · Q36 · 4 marks