Chapter 6: Triangles
CBSE Official PYQs (2015-2026)
Q1. In \(\Delta ABC\), \(D\) and \(E\) are points on sides \(AB\) and \(AC\) respectively such that \(DE \parallel BC\). If \(AD = 3\text{ cm}\), \(DB = 4\text{ cm}\) and \(AE = 6\text{ cm}\), then the length of \(EC\) is :
Q2. If \(\Delta ABC \sim \Delta PQR\) with \(\frac{BC}{QR} = \frac{1}{3}\), then \(\frac{\text{area}(\Delta PQR)}{\text{area}(\Delta ABC)}\) is equal to :
Q3. In \(\Delta ABC\) and \(\Delta DEF\), \(\angle B = \angle E\), \(\angle F = \angle C\) and \(AB = 3 DE\). Then the two triangles are :
Q4. In \(\Delta ABC\), \(P\) and \(Q\) are points on sides \(AB\) and \(AC\) respectively such that \(PQ \parallel BC\). If \(AP = 2.4\text{ cm}\), \(AQ = 2\text{ cm}\), \(QC = 3\text{ cm}\), then the length of \(AB\) is :
Q5. If in two triangles \(ABC\) and \(PQR\), \(\frac{AB}{QR} = \frac{BC}{PR} = \frac{CA}{PQ}\), then :
Q6. A girl of height 90 cm is walking away from the base of a lamp-post at a speed of 1.2 m/s. If the lamp is 3.6 m above the ground, then the length of her shadow after 4 seconds is :
Q7. In \(\Delta ABC\), \(DE \parallel BC\) such that \(AD = x\), \(DB = x-2\), \(AE = x+2\) and \(EC = x-1\). The value of \(x\) is :
Q8. In \(\Delta ABC\), \(\angle BAC = 90^\circ\) and \(AD \perp BC\) where \(D\) is a point on \(BC\). Then :
Q9. If \(\Delta ABC \sim \Delta DEF\), \(AB = 4\text{ cm}\), \(DE = 6\text{ cm}\), \(EF = 9\text{ cm}\) and \(FD = 12\text{ cm}\), then the perimeter of \(\Delta ABC\) is :
Q10. In \(\Delta ABC\), \(X\) and \(Y\) are points on sides \(AB\) and \(AC\) respectively such that \(XY \parallel BC\). If \(AX = 1\text{ cm}\), \(XB = 2\text{ cm}\), then \(\text{area}(\Delta AXY) : \text{area}(\Delta ABC)\) is :
Q11. \(P\) and \(Q\) are points on the sides \(AB\) and \(AC\) respectively of a \(\Delta ABC\) such that \(PQ \parallel BC\) and divides \(\Delta ABC\) into two parts, equal in area. The ratio of \(AP\) to \(AB\) is :
Q12. Given that \(\Delta ABC \sim \Delta APQ\), if \(BC = 8\text{ cm}\), \(PQ = 4\text{ cm}\), \(AP = 2.8\text{ cm}\) and \(AQ = 3.8\text{ cm}\), then the length of \(AC\) is :
Q13. If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. This theorem is known as :
Q14. In \(\Delta ABC\) and \(\Delta PQR\), \(\frac{AB}{QR} = \frac{BC}{RP}\). To make these triangles similar, the additional condition required is :
Q15. In \(\Delta ABC\), \(AD\) is the internal bisector of \(\angle A\) meeting \(BC\) at \(D\). If \(AB = 6\text{ cm}\), \(AC = 8\text{ cm}\) and \(BD = 3\text{ cm}\), then the length of \(CD\) is :
Q16. It is given that \(\Delta ABC \sim \Delta DFE\). If \(\angle A = 30^\circ\), \(\angle C = 50^\circ\), \(AB = 5\text{ cm}\), \(AC = 8\text{ cm}\) and \(DF = 7.5\text{ cm}\), then which of the following is true?
Q17. In \(\Delta ABC\), \(D\) and \(E\) are points on sides \(AB\) and \(AC\) respectively such that \(DE \parallel BC\) and \(AD : DB = 2 : 3\). If the area of \(\Delta ADE\) is \(20\text{ cm}^2\), then the area of \(\Delta ABC\) is :
Q18. In two similar triangles, the ratio of their corresponding sides is \(3:5\). The ratio of their corresponding altitudes is :
Q19. In \(\Delta ABC\), \(D\) and \(E\) are points on sides \(AB\) and \(AC\) respectively such that \(DE \parallel BC\) and \(AD = 2.4\text{ cm}\), \(AE = 3.2\text{ cm}\), \(EC = 4.8\text{ cm}\). The length of \(AB\) is :
Q20. In \(\Delta OAB\), \(M\) and \(N\) are points on sides \(OA\) and \(OB\) respectively such that \(MN \parallel AB\). If \(OM = 3\text{ cm}\), \(MA = 4\text{ cm}\) and \(ON = 4.5\text{ cm}\), then the length of \(NB\) is :
Q21. If the corresponding angles of two triangles are equal, then the two triangles are similar by which similarity criterion?
Q22. In \(\Delta ABC\), \(D\) and \(E\) are points on sides \(AB\) and \(AC\) respectively such that \(\angle ADE = \angle B\). If \(AD = 3.8\text{ cm}\), \(AE = 3.6\text{ cm}\), \(BE = 2.1\text{ cm}\) and \(BC = 4.2\text{ cm}\), then the length of \(DE\) is :
Q23. Corresponding sides of two similar triangles are in the ratio \(1:3\). If the area of the smaller triangle is \(15\text{ cm}^2\), then the area of the larger triangle is :
Q24. Two line segments \(QA\) and \(PB\) are perpendicular to a line segment \(AB\) on opposite sides of \(AB\). The line segment joining \(P\) and \(Q\) intersects \(AB\) at \(O\). If \(AO = 10\text{ cm}\), \(BO = 6\text{ cm}\) and \(PB = 9\text{ cm}\), then the length of \(AQ\) is :
Q25. In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This is known as :
Q26. In \(\Delta ABC\), \(D\) and \(E\) are points on the sides \(AB\) and \(AC\) respectively such that \(DE \parallel BC\). If \(AD = x\), \(DB = x+1\), \(AE = x+3\) and \(EC = x+5\), then the value of \(x\) is :
Q27. If \(\Delta ABC \sim \Delta PQR\), and the perimeters of \(\Delta ABC\) and \(\Delta PQR\) are \(36\text{ cm}\) and \(24\text{ cm}\) respectively. If \(PQ = 10\text{ cm}\), then the length of \(AB\) is :
Q28. In \(\Delta PQR\), \(S\) and \(T\) are points on sides \(PQ\) and \(PR\) respectively such that \(ST \parallel QR\). If \(PS = x\), \(SQ = 7.2\text{ cm}\), \(PT = 1.8\text{ cm}\) and \(TR = 5.4\text{ cm}\), then the value of \(x\) is :
Q29. In \(\Delta ABC\), \(\angle A = 90^\circ\), \(AB = 6\text{ cm}\) and \(AC = 8\text{ cm}\). If \(AD \perp BC\), then the length of \(AD\) is :
Q30. If \(\Delta ABC \sim \Delta PQR\), \(\frac{AB}{PQ} = \frac{2}{5}\) and the area of \(\Delta PQR\) is \(125\text{ cm}^2\), then the area of \(\Delta ABC\) is :
Q31. In a \(\Delta ABC\), let \(O\) be a point inside it. Points \(E, F\), and \(G\) lie on \(OA, OB\), and \(OC\) respectively. If \(EF \parallel AB\) and \(FG \parallel BC\), prove that \(EG \parallel AC\).
Q32. In a \(\Delta ABC\), \(D\) and \(E\) are points on the sides \(AB\) and \(AC\) respectively such that \(DE \parallel BC\). If \(AD = 2x-1\), \(DB = x-3\), \(AE = 2x+5\) and \(EC = x-1\), find the value of \(x\).
Q33. If \(\Delta ABC \sim \Delta DEF\) such that \(AB = 1.2\text{ cm}\) and \(DE = 1.4\text{ cm}\). Find the ratio of the areas of \(\Delta ABC\) and \(\Delta DEF\).
Q34. Two line segments \(AB\) and \(CD\) intersect at point \(O\) such that \(OA \cdot OB = OC \cdot OD\). Show that \(\angle A = \angle C\) and \(\angle B = \angle D\).
Q35. A vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same time a tower casts a shadow 28 m long. Find the height of the tower.
Q36. In \(\Delta PQR\), \(S\) and \(T\) are points on sides \(PR\) and \(PQ\) respectively such that \(\angle PQR = \angle PRQ\). If points \(N\) and \(M\) lie on \(PQ\) and \(PR\) respectively and \(\Delta NSQ \cong \Delta MTR\), then prove that \(\Delta PTS \sim \Delta PRQ\).
Q37. If the perimeters of two similar triangles \(ABC\) and \(PQR\) are \(30\text{ cm}\) and \(20\text{ cm}\) respectively, and one side of \(\Delta PQR\) is \(12\text{ cm}\), find the corresponding side of \(\Delta ABC\).
Q38. In \(\Delta ABC\), let \(O\) be any point inside it. \(L, M\), and \(N\) are points on \(OA, OB\), and \(OC\) respectively. If \(LM \parallel AB\) and \(LN \parallel AC\), prove that \(MN \parallel BC\).
Q39. In a similar triangle pair, \(\Delta ABC \sim \Delta PQR\). If \(AD\) and \(PM\) are medians of \(\Delta ABC\) and \(\Delta PQR\) respectively, show that \(\frac{AB}{PQ} = \frac{AD}{PM}\).
Q40. In \(\Delta ABC\), \(D\), \(E\) and \(F\) are midpoints of sides \(AB\), \(BC\) and \(CA\) respectively. Find the ratio of the area of \(\Delta DEF\) to the area of \(\Delta ABC\).
Q41. State and prove Basic Proportionality Theorem (Thales Theorem).
Q42. In \(\Delta TQR\), \(P\) is a point on \(TR\) and \(S\) is a point on \(QR\). If \(\frac{QR}{QS} = \frac{QT}{PR}\) and \(\angle PQR = \angle PRQ\), show that \(\Delta PQS \sim \Delta TQR\).
Q43. In \(\Delta ABC\), \(AD \perp BC\) such that \(AD^2 = BD \cdot CD\). Prove that \(\Delta ABC\) is a right-angled triangle at \(A\).
Q44. Let \(OB\) be the perpendicular bisector of a line segment \(DE\), intersecting at \(O\). Let \(F\) be a point such that \(FA \perp OB\), and \(FE\) intersects \(OB\) at \(C\). Prove that \(\frac{1}{OA} + \frac{1}{OB} = \frac{2}{OC}\).
Q45. Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding altitudes.
Q46. \(ABCD\) is a trapezium with \(AB \parallel DC\). Diagonals \(AC\) and \(BD\) intersect each other at \(O\). Show that \(\frac{AO}{BO} = \frac{CO}{DO}\).
Q47. In \(\Delta ABC\), \(D\) is a point on side \(BC\) such that \(\angle ADC = \angle BAC\). Prove that \(CA^2 = CB \cdot CD\).
Q48. \(AD\) and \(PM\) are altitudes of similar triangles \(ABC\) and \(PQR\) respectively. Show that \(\frac{AB}{PQ} = \frac{AD}{PM}\).
Q49. Prove that 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. Using this theorem, find the value of \(x\) if \(DE \parallel BC\) in a \(\Delta ABC\) where \(AD = 4x-3\), \(DB = 3x-1\), \(AE = 8x-7\) and \(EC = 5x-3\).
Q50. CASE STUDY: Scale Factor and Similar Figures
A scale drawing of a building or object is a drawing which has the same shape as the real object but a different size. The scale factor is the ratio of the length of a side of the drawing to the corresponding length of the real object.
A student of Class X is making a model of a historical monument (like Taj Mahal) using similarity concepts.
Based on the above information, answer the following questions:
- If two similar triangles have a scale factor of \(3:4\), what is the ratio of their perimeters? (1 Mark)
- If the area of the smaller triangle is \(36\text{ cm}^2\), what is the area of the larger triangle? (2 Marks)
- If the ratio of sides is \(k\), show that the ratio of their corresponding medians is also \(k\). (2 Marks)
1. Ratio is \(3:4\).
2. Area is \(64\text{ cm}^2\) (Since ratio of areas is \(9/16\)).
3. Standard geometric proof using SAS similarity.
Q51. CASE STUDY: Shadow and Similar Triangles
Anish and his father went to a nearby park. Anish noticed a tall electric pole and wanted to find its height. He stands at a distance of 3.2 m from the base of the pole. The length of his shadow on the ground is 1.6 m. Anish's height is 1.5 m.
Based on the above information, answer the following questions:
- Name the mathematical concept of triangles used here to find the height of the pole. (1 Mark)
- What is the height of the electric pole? (2 Marks)
- If Anish walks 1 m closer to the pole, what will be the new length of his shadow? (2 Marks)
1. Similarity of Triangles.
2. 4.5 m.
3. 1.1 m (approx).
Q52. CASE STUDY: Maps and Similarity
A cartographer is designing a map of a triangular region. The actual boundary lengths of the region are 150 km, 200 km, and 250 km. On the map, the longest side of this region is represented by a line segment of length 25 cm.
Based on the above information, answer the following questions:
- What is the scale factor used by the cartographer? (1 Mark)
- What are the lengths of the other two sides on the map? (2 Marks)
- Find the ratio of the area of the actual region to the area of the region represented on the map. (2 Marks)
1. Scale factor = \(1 : 1,000,000\) (or \(1\text{ cm} : 10\text{ km}\)).
2. 15 cm and 20 cm.
3. \(100,000,000 : 1\) (or \(10^{10} : 1\) depending on units).