1. 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?

2. 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.

3. Sathi has drawn a right-angled triangle where the length of the hypotenuse is 6 cm more than twice the length of the smallest side. If the length of the third side is 2 cm less than the hypotenuse, then calculate and write the lengths of all three sides of the triangle.

4. 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.

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

6. 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.

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

8. The radius, height, and slant height of a right circular cone always form the three sides of a right-angled triangle.

9. 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.

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

11. The height, radius, and slant height of a right circular cone always form the three sides of a right-angled triangle.

12. The height, radius, and slant height of a right circular cone always form the three sides of a right-angled triangle.

13. In a right-angled triangle, the lengths of the two sides adjacent to the right angle are 4 cm and 3 cm. If the triangle is rotated once completely about the longer of these two sides as the axis, the solid formed is a cone. Calculate and write the lateral surface area, total surface area, and volume of the cone formed.

14. In right-angled triangle ABC, where ∠B = 90°, ∠A = 30°, and AC = 20 cm, find the lengths of sides BC and AB.

15. In right-angled triangle PQR, ∠Q = 90°, ∠R = 45°, and PR = \(3\sqrt{2}\) meters. Find the lengths of sides PQ and QR.

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

17. "The height, radius, and slant height of a right circular cone always form the three sides of a right-angled triangle."

18. The three sides of a triangle are 6 cm, 8 cm, and 10 cm respectively. What is the circumradius of the triangle? This triangle is special—it’s a right triangle (since \(6^2 + 8^2 = 36 + 64 = 100 = 10^2\)). And for right triangles, the circumradius is half the hypotenuse. So the circumradius = \(\frac{10}{2} = 5\) cm.

19. In a right-angled quadrilateral, if the number of vertices is denoted by \(x\), the number of sides by \(y\), and the number of diagonals by \(z\), then what is the value of \(x + 3y - 5z\)?

(a) 14 (b) 44 (c) 20 (d) 24

20. 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.

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

22. 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.)

23. 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.)

24. 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.

25. 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.

26. 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

27. A right-angled isosceles triangle in which each of the equal sides has a length of \(a\) units. The perimeter of the triangle is —

(a) \((1+\sqrt{2})a\) units (b) \((2+\sqrt{2})a\) units (c) \(3a\) units (d) \((3+2\sqrt{2})a\) units

28. 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.)

29. Draw a right-angled triangle whose two sides adjacent to the right angle are 8 cm and 6 cm respectively, and draw an incircle of the triangle. (Only construction marks are required)

30. 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.)