1. 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.

2. 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.

3. 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.

4. "The differences between the semi-perimeter of a triangle and the lengths of its sides are respectively 4 cm, 7 cm, and 5 cm. The area of the triangle is –"

(a) \(20\sqrt{7}\) square cm (b) \(10\sqrt{14}\) square cm (c) \(20\sqrt{14}\) square cm (d) 140 square cm

5. 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)

6. The lengths of two sides are 7.6 cm and 6 cm, and the included angle between them is 75°. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).

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

8. ABC is a right-angled triangle with \(\angle B\) being the right angle and BD ⟂ AC. If AD = 4 cm and CD = 16 cm, then calculate and write the lengths of BD and AB.

9. Assume \(PQ\) is the tower and \(AB\) is the stick. \(∴ BC = 4\) cm, \(QC = 28\) meters Triangles \(∆PQC\) and \(∆ABC\) are similar right-angled triangles. Therefore, they are similar. \(∴ \cfrac{PQ}{AB} = \cfrac{QC}{BC} \) Or, \(\cfrac{PQ}{6} = \cfrac{28}{4}\) Or, \(PQ = \cfrac{28 \times 6}{4} = 42\) meters ∴ The height of the tower is 42 meters.

10. 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

11. 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

12. 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.

13. Draw a triangle in which two sides are 9 cm and 7 cm, and the included angle between them is 60°. Then draw the incircle of that triangle. (Only construction marks are required.)

14. 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.

15. The lengths of the two equal sides of an isosceles triangle are 5 cm, and the base is 6 cm. The area of the triangle is —

(a) 18 square cm (b) 12 square cm (c) 15 square cm (d) 30 square cm

16. The side lengths of two right-angled trapezoids are 4 cm, 12 cm, 15 cm, 6 cm, (2h - 1) cm, and 16 cm, respectively. If the areas of the two trapezoids are equal, find the value of h.

17. 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.

18. In a circle centered at O, there are two parallel chords AB and CD with lengths 10 cm and 24 cm, positioned on opposite sides of the center. If the distance between the chords AB and CD is 17 cm, then calculate and write the radius of the circle.

19. 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.

20. In right-angled triangle ABC, ∠B is the right angle. If AB = \(8\sqrt{3}\) cm and BC = 8 cm, then calculate the values of ∠ACB and ∠BAC.

21. I have drawn a right-angled triangle ABC in which the hypotenuse AB = 10 cm, the base BC = 8 cm, and the perpendicular AC = 6 cm. Find the sine and tangent values of angle ∠ABC.

22. The perimeters of triangles ∆ABC and ∆DEF are 30 cm and 18 cm respectively. Given that ∆ABC ~ ∆DEF, and BC and EF are corresponding sides, if BC = 9 cm, then EF = _____________ cm.

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

26. 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

27. 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

28. 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

29. 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

30. What is the radius of the circumcircle of a triangle whose sides are 20 cm, 21 cm, and 29 cm?

(a) 14\(\frac{1}{2}\) cm (b) 14 cm (c) 10 cm (d) 11\(\frac{1}{2}\) cm