1. If the three sides of a triangle are in the ratio 3:4:5, then the triangle will always be a right-angled triangle.

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

3. 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.

4. If the ratio of the three sides of a triangle is 3:4:5, then the triangle will be a right-angled triangle.

5. \(\triangle\)ABC ~ \(\triangle\)DEF; BC and EF are corresponding sides. If BE : EF = 1 : 3, then the ratio of the areas of \(\triangle\)ABC and \(\triangle\)DEF will be 1 : 27.

6. If the angles of a triangle are in the ratio \(1 : 1 : 2\), then what will be the ratio of the sides of the triangle?

(a) \(1:\sqrt{2}:1 \) (b) \(1:1:\sqrt{2}\) (c) \(1:1:2\) (d) \(1:2:1\)

7. If the angles of a triangle are in the ratio 1:1:2, then the sides of the triangle will be in the ratio ———.

8. 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?

9. Draw a right-angled triangle whose hypotenuse is 10 cm and one of the other sides is 6.5 cm. Then, draw the incircle of this triangle. (Only construction marks are required.)

10. Draw a right-angled triangle whose hypotenuse is 9 cm and one of the other sides is 5.5 cm. Then draw an incircle of the triangle. (Only the construction marks are required.)

11. Draw a right-angled triangle whose two sides adjacent to the right angle are 4.5 cm and 6 cm. Then draw the incircle of that triangle. (Only construction marks are required.)

12. If the three consecutive angles of a cyclic quadrilateral are in the ratio \(1:2:5\), then the measure of the fourth angle will be _____.

13. A line parallel to side BC of triangle ABC intersects sides AB and AC at points D and E, respectively. If AD is half of AE, then the ratio BD:EC will be –

(a) 1:2 (b) 2:1 (c) 2:3 (d) 1:3

14. Draw a right-angled triangle where the two sides adjacent to the right angle are 5 cm and 6 cm. Then, draw an incircle (a circle inscribed inside the triangle) within that triangle.

15. In a right-angled triangle, if the ratio of the perpendicular (opposite side) to the hypotenuse with respect to a positive acute angle \(\theta\) is \(12 : 13\), then determine the ratio of the perpendicular to the base and the ratio of the hypotenuse to the base, and verify that \( \sec^2\theta = 1 + \tan^2\theta \).

16. Match the items on the left with those on the right (any two):

17. If the ratio of the capital investments of A, B, and C is \(\cfrac{1}{2} : \cfrac{1}{3} : \cfrac{1}{4}\) and the total profit is ₹520, then C's share of the profit will be

(a) ₹120 (b) ₹160 (c) ₹180 (d) ₹140

18. If the ratio of the radii of two vertical solid circular cylinders is 2:3, and the ratio of their heights is 5:3, then the ratio of their curved surface areas will be –

(a) 2:5 (b) 8:7 (c) 10:9 (d) 16:9

19. 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

20. "If the two acute angles of a right-angled triangle are in the ratio 2:3, what are the radian measures of those two angles?

(a) \(\cfrac{π}{5},\cfrac{3π}{10}\) (b) \(\cfrac{π}{10},\cfrac{3π}{5}\) (c) \(\cfrac{π}{5},\cfrac{3π}{20}\) (d) \(\cfrac{π}{5},\cfrac{π}{15}\)

21. 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) \(\cfrac{10}{13}\)

22. If the radii of two cones are in the ratio 2:3 and their heights are in the ratio 5:3, then what will be the ratio of their volumes?

(a) 4:9 (b) 9:4 (c) 27:20 (d) 20:27

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

24. If the three angles of a triangle are in the ratio 2:3:4, then the measure of the largest angle in degrees is ________.

25. Draw a right-angled triangle whose two sides adjacent to the right angle are 7 cm and 9 cm respectively. Then draw the incircle of that triangle. (Only construction marks are required.)

26. A solid lead sphere with a diameter of 12 cm is melted down to form three smaller solid spheres. If the diameters of the smaller spheres are in the ratio 3 : 4 : 5, then what is the radius of each of the smaller spheres?

27. 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?

28. In a right-angled triangle, the hypotenuse is 6 cm longer than one of the other two sides and 12 cm longer than the other. Find the area of the triangle.

29. In triangle \(\triangle ABC\), the incircle touches the sides AB, BC, and CA at points D, E, and F respectively. Given: AD = 12 cm, BE = 5 cm, and CF = 4 cm. Find the lengths of AB, BC, and CA.

30. If the sides of a triangle are in the ratio \(3x : 4x : 5x\), what is the measure of the largest angle of the triangle?

(a) 105\(^o\) (b) 75\(^o\) (c) 60\(^o\) (d) None of the above