1. Find the radius of the circumcircle of a triangle whose sides are 3 cm, 4 cm, and 5 cm.

2. 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).

3. Draw a triangle whose two sides are 7 cm and 6 cm, and the included angle is 75°. Then draw the incircle of the triangle.

4. 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.

5. 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

6. In triangle ABC, \(\angle\)ABC is a right angle, and AB = 5 cm, BC = 12 cm. What is the radius of the circumcircle of triangle ABC?

7. In triangle \(\triangle ABC\), AB = AC. Points E and F are the midpoints of sides AB and AC respectively. AD is perpendicular to BC, and AD = 4 cm. If EF = 3 cm, then what is the length of BD?

(a) 4 cm (b) 3 cm (c) 6 cm (d) 7 cm

8. In triangle ABC, E and F are the midpoints of sides AB and AC respectively. If the area of triangle AEF is 50 square cm, then what is the area of triangle ABC?

(a) 100 sq cm (b) 200 sq cm (c) 150 sq cm (d) 300 sq cm

9. 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.)

10. In triangle △ABC, if ∠ABC = 90°, AB = 5 cm, and BC = 12 cm, then what is the radius of its circumcircle?

11. 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

12. "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

13. Draw a triangle whose sides are 7 cm, 6 cm, and 5.5 cm respectively. Draw the incircle of the triangle. (Only provide the diagram)

14. The perimeters of two similar triangles are 20 cm and 16 cm respectively. If the length of a side of the first triangle is 4 cm, then the length of the corresponding side of the second triangle will be ____ .

15. In a circle centered at \(O\), two chords \(AB\) and \(CD\) have equal lengths. Given that \(\angle AOB = 60°\) and the radius of the circle is \(6\) cm, find the area of \(\triangle COD\).

(a) \(9\sqrt3\) square c.m (b) \(6\sqrt3\) square c.m (c) \(2\sqrt3\) square c.m (d) \(3\sqrt3\) square c.m

16. In a circle with a radius of 5 cm, AB and AC are two equal chords. The center of the circle is located outside the triangle ABC. If AB = AC = 6 cm, determine the length of the chord BC.

17. In a circle centered at \(O\), two equal chords \(AB\) and \(CD\) are given. If \(\angle AOB = 60^\circ\) and the radius of the circle is 6 cm, determine the area of \( \triangle COD \).

(a) \(6\sqrt3\) square cm (b) \(2\sqrt3\) square cm (c) \(2\sqrt3\) square cm (d) \(9\sqrt3\) square cm

18. If the height and diameter of a right circular cylinder are equal, and its volume is 2156 cubic cm, then find the radius.

(a) 5 cm (b) 6 cm (c) 7 cm (d) 9 cm

19. The perimeters of two similar triangles are 20 cm and 16 cm respectively. If one side of the first triangle is 9 cm, what is the length of the corresponding side of the second triangle?

20. “The perimeters of two similar triangles are 20 cm and 16 cm respectively. If the length of one side of the first triangle is 9 cm, what is the length of the corresponding side of the second triangle.”

21. 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.

22. When a spherical gas balloon is inflated and its radius increases from 7 cm to 21 cm, determine the ratio of its surface areas before and after inflation.

23. The length of one side is 6.2 cm, and the measures of the two angles adjacent to that side are 50° and 75°. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).

24. In isosceles triangle ABC, the circumcenter is O and \(\angle ABC = 120^\circ\). If the radius of the circle is 5 cm, find the length of side AB.

25. “The perimeters of two similar triangles are 20 cm and 16 cm respectively. If the length of one side of the first triangle is 9 cm, calculate the length of the corresponding side of the second triangle.”

26. 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.

27. If the base radius of a right circular cone is 3 cm and the height is 4 cm, then the lateral surface area of the cone will be.

(a) \(10\pi \, cm^2\) (b) \(15\pi \, cm^2\) (c) \(12\pi \, cm^2\) (d) \(18\pi \, cm^2\)

28. 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

29. 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

30. In triangle ABC, \(\angle\)BAC = 90° and AD is perpendicular to BC. Given: AD = 8 cm, BC = 20 cm, and CD > BD Find: The length of CD.

(a) 6 cm (b) 4 cm (c) 20 cm (d) 16 cm