1. In triangle \( \triangle ABC \), a straight line parallel to side BC intersects sides AB and AC at points P and Q respectively. Given that \( AP = 18 \) cm, \( QC = 9 \) cm, and \( AQ = 2 \times PB \), find the length of \( PB \). (a) 6 cm (b) 12 cm (c) 18 cm (d) 9 cm

2. On triangle \( \triangle ABC \), points P and Q are such that \( \angle ABC = \angle APQ \). Given that \( AP = 3.6 \) cm, \( QC = 1.6 \) cm, and \( AQ = 4.8 \) cm, find the length of \( PB \). (a) 1.2 cm (b) 2.4 cm (c) 6 cm (d) None of the above

3. In trapezium ABCD, AD is parallel to BC. A straight line parallel to BC intersects AB and DC at points P and Q respectively. If \( AP : PB = 2 : 1 \), then what is the ratio \( DQ : QC \)? (a) 1:1 (b) 1:2 (c) 1:4 (d) 2:1

4. In triangle \( \triangle ABC \), AD is a median. Point E divides AD in the ratio 1:2. The extended line BE intersects AC at point F. If \( AC = 10 \) cm, find the length of \( AF \). (a) 5 cm (b) 4 cm (c) 2 cm (d) None of the above

5. In triangle \( \triangle ABC \), a line parallel to side BC intersects sides AB and AC at points D and E respectively. If \( AB = 20 \) cm and \( BD = 14 \) cm, then what is the ratio \( DE : BC \)? (a) 7:10 (b) 5:17 (c) 3:10 (d) 7:17

6. In triangle \( \triangle ABC \), a line parallel to side BC intersects sides AB and AC at points P and Q respectively. If \( AB = 3 \times PB \) and \( BC = 18 \) cm, then what is the length of \( PQ \)? (a) 10 cm (b) 9 cm (c) 12 cm (d) 8 cm

7. If a triangle similar to one with sides 4 cm, 6 cm, and 8 cm has its largest side measuring 6 cm, what is the length of the smallest side of that triangle? (a) 4 cm (b) 3 cm (c) 2 cm (d) 5 cm

8. In triangle \( \triangle ABC \), AD is a median. A straight line parallel to side BC intersects sides AB, AD, and AC at points P, O, and Q respectively. What is the ratio \( PO : OQ \)? (a) 1:2 (b) 2:3 (c) 1:1 (d) None of the above

9. AB and CD are two parallel straight lines. AD and BC intersect each other at point O. If OA = 2 cm, OB = 3 cm, and OD = 4 cm, then what is the length of OC? (a) 6 cm (b) 4 cm (c) 4.8 cm (d) 4.2 cm

10. "Line segments AB and PQ intersect at point O. AP and BQ are perpendiculars to AB. OA = 20 cm, OB = 8 cm, AP = 10 cm. Find the length of BQ." (a) 4 cm (b) 6 cm (c) 8 cm (d) None of the above

11. In trapezium ABCD, AB \(\parallel\) DC and the diagonals AC and BD intersect at point O. Given: OA = 2 × OC and AB = 10 cm. Find: The length of DC. (a) 4 cm (b) 5 cm (c) 6 cm (d) 8 cm

12. In triangle ABC, \(\angle\)BAC = 90° and AD is perpendicular to BC. Given: AD = 8 cm, BC = 20 cm, and CD > BD Find: The length of CD. (a) 6 cm (b) 4 cm (c) 20 cm (d) 16 cm

13. In triangle ABC, \(\angle\)BAC = 90° and AD is perpendicular to BC. Given: AB : AC = 3 : 4 Find: What is the ratio BD : DC? (a) 3:4 (b) 9:16 (c) 2:3 (d) None of the above

14. In triangle ABC, AB = AC and E, F are the midpoints of AB and AC respectively. AD is perpendicular to BC. Given: AD = \(2\sqrt{5}\) cm and EF = 4 cm Find: The length of AB. (a) 7 cm (b) 4 cm (c) 6 cm (d) 5 cm

15. ABC and POR are two similar triangles. If BC = 5 cm, QR = 4 cm, and the height AD = 3 cm, then what is the length of the height PE? (a) 4.2 cm (b) 1.25 cm (c) 5.4 cm (d) 2.4 cm

16. If in triangle \( \triangle ABC \), the sum of the medians AD, BE, and CF is \(x\), and the sum of the sides is \(y\), then the relationship between \(x\) and \(y\) is — (a) \(x\gt y\) (b) \(x\lt y\) (c) \(x= y\) (d) None of the above

17. In the parallelogram ABCD, point P is the midpoint of side CD. If the area of triangle \( \triangle APD \) is 25 square centimeters, then what is the area of the parallelogram? (a) 100 square cm (b) 75 square cm (c) 150 square cm (d) 50 square cm

18. If AD is the median of triangle ABC and E is any point on AD, then— (a) \(\triangle\)AEB=\(\triangle\)AEC (b) \(\triangle\)AEB<\(\triangle\)AEC (c) \(\triangle\)AEB=\(\triangle\)BED (d) \(\triangle\)AEC=\(\triangle\)DEC

19. In the adjacent figure, BX and CY are the bisectors of ∠ABC and ∠ACB respectively. Given that AB = AC and BY = 4 cm, find the length of AX. (a) 4 cm (b) 8 cm (c) 6 cm (d) 10 cm

20. In triangle ∆ABC, a straight line parallel to side BC intersects sides AB and AC at points P and Q respectively. Given that PB = AQ, AP = 9 units, and QC = 16 units, what is the length of PB? (a) 12 cm (b) 6 cm (c) 8 cm (d) 10 cm

21. In triangles \(ABC\) and \(DEF\), if \(\angle A = \angle F = 40^\circ\), \(AB:ED = AC:EF\), and \(\angle F = 65^\circ\), find the value of \(AB\). (a) 35° (b) 65° (c) 75° (d) 85°

22. In triangle \(\triangle ABC\), AB = AC. Points E and F are the midpoints of sides AB and AC respectively. AD is perpendicular to BC, and AD = 4 cm. If EF = 3 cm, then what is the length of BD? (a) 4 cm (b) 3 cm (c) 6 cm (d) 7 cm

23. In triangle \(\triangle ABC\), D and E are the midpoints of sides AB and AC respectively. If the area of triangle ABC is 36 square centimeters, then what is the area of triangle ADE? (a) 6 square centimeters (b) 9 square centimeters (c) 12 square centimeters (d) 16 square centimeters

24. If \(\angle\)A = 65° in parallelogram ABCD, then the measures of \(\angle\)B, \(\angle\)C, and \(\angle\)D respectively are: (a) 65°, 115°, 115° (b) 115°, 115°, 65° (c) 115°, 65°, 115° (d) 65°, 65°, 115°

25. If E and F are the midpoints of sides AB and AD respectively in parallelogram ABCD, and the area of parallelogram ABCD is 1600 square centimeters, then what is the area of triangle AEF? (a) 400 sq cm (b) 200 sq cm (c) 300 sq cm (d) None of the above

26. The bisectors of \(\angle\)A and \(\angle\)B of parallelogram ABCD intersect at point O. What is the measure of \(\angle\)AOB? (a) 30° (b) 60° (c) 90° (d) 45°

27. A straight line parallel to side BC of \(\triangle\)ABC intersects AB and AC at points P and Q, respectively. If AQ = 2AP, then what is the ratio PB:QC? (a) 1:2 (b) 2:1 (c) 1:1 (d) None of these

28. If the quadrilateral formed by joining the midpoints of the sides of parallelogram ABCD has an area of 100 square cm, then what is the area of the parallelogram ABCD? (a) 400 sq cm (b) 200 sq cm (c) 600 sq cm (d) 800 sq cm

29. In triangle ABC, E and F are the midpoints of sides AB and AC respectively. If the area of triangle AEF is 50 square cm, then what is the area of triangle ABC? (a) 100 sq cm (b) 200 sq cm (c) 150 sq cm (d) 300 sq cm

30. In triangle ABC, a straight line parallel to side BC intersects sides AB and AC at points P and Q respectively. AP = QC, AB = 12 cm, AQ = 2 cm. Find the length of CQ. (a) 4 cm (b) 6 cm (c) 9 cm (d) None of the above

31. In triangle \( \triangle ABC \), a straight line parallel to side BC intersects sides AB and AC at points P and Q respectively. Given that \( AP : PB = 2 : 1 \) and \( AC = 18 \) cm, find the length of \( AQ \). (a) 12 cm (b) 9 cm (c) 6 cm (d) None of the above

32. In right-angled triangle \( \triangle ABC \), where \( \angle A = 90^\circ \), a perpendicular \( AD \) is drawn from point \( A \) to the hypotenuse \( BC \). Prove that: \[ \frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle ACD} = \frac{BC^2}{AC^2} \]

33. দুটি সদৃশ ত্রিভুজের পরিসীমা যথাক্রমে 20 সেমি ও 16 সেমি। প্রথম ত্রিভুজের একটি বাহুর দৈর্ঘ্য 9 সেমি হলে, দ্বিতীয় ত্রিভুজের অনুরূপ বাহুর দৈর্ঘ্য কত? - translate in english

34. Two acute-angled triangles ∆ABC and ∆PQR are similar. Their circumcenters are X and Y respectively. If BC and QR are corresponding (similar) sides, then prove that BX : QY = BC : QR.

35. The corresponding sides of two similar triangles are _____.

36. Prove that a straight line which divides two sides of a triangle in the same ratio is parallel to the third side.

37. If the lengths of the sides of two triangles are in proportion, then the triangles will be ——.

38. POR is a triangle. A line is drawn parallel to side QR through point X, the midpoint of PQ, and it intersects side PR at point Y. Prove that point Y is the midpoint of PR.

39. The perimeters of two similar triangles are 27 cm and 16 cm respectively. If one side of the first triangle is 9 cm, then find the length of the corresponding side of the second triangle.

40. In triangle ABC, a straight line parallel to side BC intersects AB at point P and AC at point Q. If AP = 4 cm, QC = 9 cm, and PB = AQ, then find the length of PB.

41. If a perpendicular is drawn from the right angle vertex of a right-angled triangle to the hypotenuse, then the two triangles formed on either side of that perpendicular are similar to each other. — Prove it.

42. ABCD is a trapezium where AB \(\parallel\) CD. The diagonals AC and BD intersect at point O. Prove that: AO × OD = BO × OC.

43. In triangle ABC, a straight line parallel to side BC intersects AB and AC at points D and E respectively. If AE = 2AD, then find the ratio DB : EC.

44. If the bases of two triangles lie on the same straight line and the other vertex of both triangles is common, then the ratio of their areas is ______to the ratio of the lengths of their bases.

45. Two triangles will be similar if their corresponding sides are _____.

46. In triangle ABC, DE || BC, where D and E lie on sides AB and AC respectively. If AD = 5 cm, DB = 6 cm, and AE = 7.5 cm, then find the length of AC.

47. In trapezium ABCD, BC \(\parallel\) AD and AD = 4 cm. The diagonals AC and BD intersect at point O in such a way that \(\frac{AO}{OC} = \frac{DO}{OB} = \frac{1}{2}\). Find the length of BC.

48. In triangle △ABC, ∠ABC = 90° and BD ⊥ AC. If AB = 5 cm and BC = 12 cm, then what is the length of BD?

49. Prove that the line segment joining the midpoints of two sides of a triangle is equal to half of the third side.

50. If two triangles are similar, prove that their corresponding sides are proportional.

51. PQRS is a cyclic quadrilateral in which side QR is extended up to point T. If the measures of angles ∠SRQ and ∠SRT are in the ratio 4:5, then find the measures of ∠SPQ and ∠SRQ.

52. In \(\triangle\)ABC, \(\angle\)ABC = 90°, and BD \(\bot\) AC. If BD = 6 cm and AD = 4 cm, then what is the length of CD?

53. If the ratio of the areas of two similar triangles is 64:49, then find the ratio of their corresponding sides.

54. In a right-angled isosceles triangle ABC, ∠B is the right angle. The bisector of ∠BAC intersects BC at point D. If BD = 2 cm, then what is the length of CD?

55. In triangle ABC, let X and Y be the midpoints of sides AB and AC respectively. If \(BC + XY = 12\) units, then what is the value of \(BC - XY\)?

56. In triangle ∆ABC, a straight line parallel to side BC intersects AB and AC at points P and Q respectively. Given that \(\frac{AQ}{QC} = \frac{3}{4}\) and AB = 21 cm, find the length of PB.

57. In triangle ∆ABC, a straight line parallel to side BC intersects sides AB and AC at points X and Y respectively. If AX = 2.4 cm, AY = 3.2 cm, and YC = 4.8 cm, find the length of AB.

58. If \( DE || BC \) and \( BD = (x - 3) \) cm, \( AB = 2x \) cm, \( CE = (x - 2) \) cm, and \( AC = (2x + 3) \) cm, determine the value of \( x \).

59. A straight line parallel to side BC of triangle ∆ABC intersects sides AB and AC at points D and E respectively. If AD : BD = 3 : 5, then what is the ratio of the area of triangle ∆ADE to the area of trapezium DBCE?

60. If two chords of a circle intersect each other, prove that the product of the segments of one chord is equal to the product of the segments of the other chord.

61. On the sides AC and BC of \(\triangle\)ABC, two points L and M are positioned respectively such that \(LM \parallel AB\), and \(AL = (x - 2)\) units, \(AC = 2x + 3\) units, \(BM = (x - 3)\) units, and \(BC = 2x\) units. Then, find the value of \(x\).

62. Given: AC is the diameter of a circle centered at O, ABC is an inscribed triangle, and OP ⊥ AB (where P lies on the circle). **Prove that:** \[ OP : BC = 1 : 2 \]

63. In trapezium ABCD, AB and DC are parallel. A straight line is drawn parallel to AB, which intersects AD and BC at points E and F respectively. Prove that: \[ AE : ED = BF : FC \]

64. Given that ∆ABC ~ ∆DEF and the corresponding sides AB, BC, and CA of ∆ABC match with DE, EF, and DF of ∆DEF respectively; If ∠A = 47° and ∠E = 83°, then what is the measure of ∠C?