1. "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) \(120\pi\)

2. ABC is a right-angled triangle with hypotenuse BC. From point A, a perpendicular AD is drawn to BC. If BD = 4 cm and DC = 5 cm, then what is the length of AB?

3. 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) \(p^2=a^2-b^2\)

4. 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

5. 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.

6. In the 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 it at point D. Find the length of BD.

7. 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?

8. 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.

9. 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: \[ \frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle ACD} = \frac{BC^2}{AC^2} \]

10. 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.

11. If the circumradius of right-angled triangle ABC is 5 cm, then the length of the hypotenuse is –

(a) 2.5 cm (b) 10 cm (c) 5 cm (d) None of the above

12. From the vertex A of triangle \( \triangle ABC \), a perpendicular AD is drawn to the side BC. If \[ \frac{BD}{DA} = \frac{DA}{DC} \] then prove that triangle ABC is a right-angled triangle.

13. Prove that if a perpendicular is drawn from the right angle vertex of a right-angled triangle to the hypotenuse, the two triangles formed are similar to each other.

14. In a right-angled triangle, the hypotenuse is 15 cm, and the difference between the other two sides is 3 cm. Find the lengths of those two sides.

15. Prove that 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 the perpendicular are similar to each other.

16. Prove that if a perpendicular is drawn from the right-angled vertex of a right-angled triangle to the hypotenuse, then the two triangles formed on either side of this perpendicular are similar, and each of these triangles is similar to the original triangle.

17. Prove that if a perpendicular is drawn from the right-angled vertex of any right-angled triangle to the hypotenuse, then the two resulting triangles on either side of the perpendicular are similar to each other and each is also similar to the original triangle.

18. Prove that if a perpendicular is drawn from the right-angled vertex of a right-angled triangle to the hypotenuse, then the two triangles formed on either side of this perpendicular are similar to each other and each is similar to the original triangle.

19. If a perpendicular is drawn from the right-angled vertex of any right triangle to the hypotenuse, then the two triangles formed on either side of this perpendicular are similar to each other, and each of them is also similar to the original triangle.

20. From vertex \(A\) of \(\triangle ABC\), a perpendicular \(AD\) is drawn to side \(BC\). If \(\frac{BD}{DA} = \frac{DA}{DC}\), then prove that \(\triangle ABC\) is a right-angled triangle.

21. From vertex A of triangle \(\triangle ABC\), a perpendicular AD is drawn to the base BC, intersecting BC at point D. If \(AD^2 = BD \cdot CD\), prove that triangle ABC is a right-angled triangle and that \(\angle A = 90^\circ\).

22. If the lengths of the three sides of a triangle are 4 cm, 15 cm, and 17 cm, then calculate and write whether the triangle is a right-angled triangle or not.

23. Amina has drawn a right-angled triangle with the two sides adjacent to the right angle measuring 15 cm and 20 cm. When the triangle is revolved once completely around the 15 cm side as the axis, it forms a solid. Calculate the lateral surface area, the total surface area, and the volume of the solid formed.

24. In right-angled triangle \( \triangle ABC \), \( \angle ABC = 90^\circ \) and BD is perpendicular to AC. If BD = 8 cm and AD = 4 cm, what is the length of CD?

25. In a right-angled triangle, the two acute angles are \(\theta\) and \(\phi\). If \( \tan\theta = \cfrac{5}{12} \), then what is the value of \( \sin\phi \)?

(a) \(\cfrac{12}{13}\) (b) \(\cfrac{5}{13}\) (c) \(\cfrac{1}{4}\) (d) \(\cfrac{10}{13}\)

26. 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

27. 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.

28. 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?

29. Perfectly explained. The triangle formed is a right-angled isosceles triangle, where the angle of elevation is 45°, making the opposite and adjacent sides equal. So, the height of the pillar equals the distance from its base: 20 meters.

(a) 1:2:3 (b) 1:5:7 (c) 1:7:9 (d) None of the above

30. If the lengths of the three sides of a triangle are in the ratio 3:4:5, then the triangle is a right-angled triangle.