1. In triangle ABC, \(\angle\)BAC = 90°, and AD is perpendicular to BC. Given: AC = 8 cm, AB = 6 cm Find: The length of BD. (a) 6 cm (b) 1.5 cm (c) 3 cm (d) 3.6 cm

2. If one diagonal of a rhombus with side length 17 cm is 16 cm, what is the length of the other diagonal? (a) 30 cm (b) 25 cm (c) 32 cm (d) 18 cm

3. Which of the following ratios could represent the lengths of the three sides of a right-angled triangle? (a) 4:5:6 (b) 5:6:7 (c) 4:6:7 (d) 3:4:5

4. If the sides of a triangle are in the ratio \(\sqrt{7} : \sqrt{3} : 2\), then what type of triangle is it? (a) right angle. (b) equilateral (c) isosceles (d) obtuse-angle

5. In right-angled triangle ABC, \(\angle B = 90^\circ\), and M is the midpoint of the hypotenuse drawn from point B. If AB = 6 and AC = 8, then what is the length of BM? (a) 10 (b) 4 (c) 16 (d) None of the above

6. The three sides of a triangle are 4 cm, 5 cm, and 7 cm respectively. Therefore, the triangle is – (a) acute-angled (b) obtuse-angled (c) right-angled (d) None of the above

7. Here’s the English translation of your math problem: In triangle \( \triangle ABC \), \( \angle ABC = 90^\circ \), \( BC = 24 \) cm, and \( E \) is the midpoint of \( AC \). If \( ED \perp BC \), then what is the length of \( BD \)? (a) 6 cm (b) 8 cm (c) 9 cm (d) None of the above

8. In the rhombus PQRS, if ∠Q = 60° and RS = 6 cm, what is the length of the diagonal PR? (a) \(6\sqrt3\) cm (b) \(\sqrt3\) cm (c) \(\cfrac{1}{\sqrt3}\) cm (d) 6 cm

9. Which of the following ratios could represent the lengths of the three sides of a right-angled triangle? (a) 4:5:6 (b) 5:6:7 (c) 4:6:7 (d) 3:4:5

10. If the length of the median drawn from the right-angled vertex of a right-angled triangle is 5 cm, what is the area of the triangle’s circumcircle? (a) 78\(\frac78\(\frac{2}{7}\) square cm{2}{7}\ (b) 78\(\frac{3}{7}\) square cm (c) 78\(\frac{5}{7}\) square cm (d) 78\(\frac{4}{7}\) suare cm

11. If the lengths of the perpendicular sides of a right-angled triangle are \(a\) and \(b\), and the length of the perpendicular drawn from the right-angled vertex to the hypotenuse is \(p\), then – (a) \( \cfrac{1}{p^2} =\cfrac{1}{a^2} +\cfrac{1}{b^2} \) (b) \( \cfrac{1}{p^2} =\cfrac{1}{a^2} -\cfrac{1}{b^2} \) (c) \(p^2=a^2+b^2\) (d) \(p^2=a^2-b^2\)

12. ABCD is a rectangle. O is the point where the diagonals intersect. If AB = 4 cm and OD = 2.5 cm, then what is the length of BC? (a) 4 cm (b) 1.5 cm (c) 3 cm (d) 2 cm

13. 2 (a) 3 (b) 4 (c) 5 (d) 6

14. If \(AC = BC\) in a triangle and \(AB^2 = 2AC^2\), then the measure of \(\angle C\) is _____. (a) 30° (b) 45° (c) 60° (d) 90°

15. In triangle ABC, ∠A is a right angle. From point A, a line is drawn to point D, the midpoint of the hypotenuse BC. If BC = 10 cm, then the length of AD is — (a) 5 cm (b) 6 cm (c) 7 cm (d) 8 cm

16. In triangle ABC, if AB\(^2\) + BC\(^2\) = AC\(^2\) and AC = \(\sqrt{2}\) × BC, then what type of triangle is it? (a) Right-angled (b) Right-angled isosceles (c) Equilateral (d) Isosceles

17. "If the length of the median drawn from the right angle vertex of a right-angled triangle is 10 cm, what is the area of the triangle's circumcircle?" (a) \(100\pi\) (b) \(25\pi\) (c) \(50\pi\) (d) \(120\pi\)

18. If the three angles of a triangle are in the ratio 3:4:5, then the triangle will always be a right-angled triangle. True / False

19. If the three sides of a triangle are in the ratio 5 : 12 : 13, then the triangle will always be a right-angled triangle. True / False

20. Prove that in a triangle, if the area of the square constructed on one side is equal to the sum of the areas of the squares constructed on the other two sides, then the angle opposite the longest side is a right angle.

21. State and prove Pythagoras' theorem.

22. Given: ∠ABC = 90° and BD ⊥ AC; BD = 6 cm and AD = 4 cm. Find the length of CD.

23. If the lengths of the three sides of a triangle are 8 cm, 15 cm, and 17 cm respectively, can the triangle be a right-angled triangle?

24. State and prove the Pythagorean theorem.

25. In triangle △ABC, if AB = \((2a - 1)\) cm, AC = \(2\sqrt{2a}\) cm, and BC = \((2a + 1)\) cm, then find the measure of ∠BAC.

26. Write and prove the Pythagorean theorem.

27. In right-angled triangle PQR, ∠P = 90°, and PS is perpendicular to the hypotenuse QR. Prove that: $$ \frac{1}{PS^2} - \frac{1}{PQ^2} = \frac{1}{PR^2} $$

28. Prove that in a right-angled triangle, the area of the square constructed on the hypotenuse is equal to the sum of the areas of the squares constructed on the other two sides.

29. If the area of the square drawn on one side of any triangle is equal to the sum of the areas of the squares drawn on the other two sides, then prove that the angle opposite to the first side is a right angle.

30. State and prove the Pythagorean theorem.

31. State and prove the Pythagorean theorem.

32. Write Pythagoras' theorem and prove it.

33. Prove that if a perpendicular is drawn from the right angle vertex of a right-angled triangle to the hypotenuse, then the two adjacent triangles formed are similar to each other and each is also similar to the original triangle.

34. In triangle ABC, ∠A = 90°, AB = 12 cm, AC = 5 cm, and BC = 13 cm. A perpendicular AD is drawn from point A to side BC. What is the length of AD?

35. State and prove Pythagoras' theorem.

36. In triangle △ABC, ∠A is a right angle and BP and CQ are medians. Prove that: \[ 5BC^2 = 4(BP^2 + CQ^2) \]

37. State and prove the converse of Pythagoras' theorem.

38. In \(\triangle\)ABC, if AB = \((2a-1)\) cm, AC = \(2\sqrt{2}a\) cm, and BC = \((2a+1)\) cm, then write the value of \(\angle\)BAC.

39. State and prove the Pythagorean theorem.

40. In right-angled triangle ABC, ∠ABC = 90°, AB = 3 cm, BC = 4 cm, and from point B, a perpendicular BD is drawn to side AC, meeting AC at point D. Find the length of BD.

41. A right-angled triangle has two sides adjacent to the right angle measuring 4 cm and 3 cm. If the triangle is rotated once completely around its hypotenuse as the axis, find the volume of the solid formed.

42. In a right-angled triangle, the lengths of the two sides adjacent to the right angle are 4 cm and 3 cm respectively. If the triangle is rotated once completely around the longer of the two adjacent sides, find the total surface area and the volume of the solid formed.

43. In triangle ABC, AB = (2a − 1) cm, AC = 2√2a cm, and BC = (2a + 1) cm. Find the measure of ∠BAC.

44. Prove that if ABCD is a rectangle and 'O' is any point inside it, then \( OA^2 + OC^2 = OB^2 + OD^2 \).

45. In \(\triangle\)ABC, if AB \(= (2p-1)\) cm, AC \(= 2\sqrt2p\) cm, and BC \(= (2p+1)\) cm, then the value of \(\angle\)BAC is...? Let me know if you need further assistance!

46. If each side of a rhombus is 5 cm and one of its diagonals is 4 cm, find the length of the other diagonal.

47. State and prove Pythagoras' theorem.

48. State and prove the Pythagorean Theorem.

49. Point O lies inside the rectangle ABCD such that OB = 6 cm, OD = 8 cm, and OA = 5 cm. Find the length of OC.

50. PORS is a rectangle, and A is any point inside the rectangle. Prove that: \[ AP^2 + AR^2 = AQ^2 + AS^2 \]

51. In triangle ABC, AD is perpendicular to side BC and \(AD^2 = BD \cdot DC\); prove that \(\angle BAC\) is a right angle.