1. If the orthocenter, centroid, circumcenter, and incenter of a triangle all lie at the same point, then what type of triangle is it? (a) isosceles (b) scalene (c) equilateral (d) right-angled isosceles

2. 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

3. The circumcenter of an acute-angled triangle lies inside the triangle. (a) inside (b) outside (c) on the hypotenuse (d) on the hypotenuse

4. If the area of the incircle of an equilateral triangle is 49 ???? cm 2 , what is the perimeter of the triangle? (a) 21\(\sqrt3\) (b) 7\(\sqrt3\) (c) 14\(\sqrt3\) (d) 42\(\sqrt3\)

5. What is the radius of the incircle of an equilateral triangle whose each side is \(2a\)? (a) \(\sqrt{3a}\) cm (b) \(\cfrac{a}{\sqrt3}\) cm (c) \(\cfrac{a}{\sqrt2}\) cm (d) \(\sqrt{2a}\) cm

6. What is the ratio of the area of an equilateral triangle to the area of its circumcircle? (a) (b) (c) (d)

7. If the circumradius of an equilateral triangle is 4 cm, what is its inradius? (a) 2 cm (b) 4 cm (c) 1 cm (d) 3 cm

8. If the circumradius of an equilateral triangle is 4 cm, what is its inradius? (a) 4 cm (b) 1.5 cm (c) 3 cm (d) 2 cm

9. The radius of the circumcircle of an equilateral triangle with side length \(5\sqrt{3}\) cm will be — (a) 6 cm (b) 5 cm (c) 4 cm (d) 3 cm

10. O is the circumcenter of triangle ABC. If \(\angle\)BAC = 85° and \(\angle\)BCA = 70°, then what is the measure of \(\angle\)OAC? (a) \(65^o\) (b) \(42\cfrac{1}{2}^o\) (c) \(50^o\) (d) \(25^o\)

11. The circumcenter of triangle \(\triangle\)ABC lies outside the triangle. If the largest angle of the triangle is \(\angle\)BAC, then: (a) \(\angle\)BAC = 90° (b) \(\angle\)BAC<90° (c) \(\angle\)BAC > 90° (d) \(\angle\)BAC = \(\angle\)ACB = \(\angle\)ABC

12. In triangle ABC, the incenter is I. When the internal bisector of ∠A (i.e., AI) is extended, it intersects the circumcircle at point P. If PB = 15 cm, then what is the length of PI? (a) 5 cm (b) 15 cm (c) 10 cm (d) 20 cm

13. In any right-angled triangle, the hypotenuse is the diameter of the circumcircle of the triangle. True / False

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

16. Draw a right-angled triangle whose two sides adjacent to the right angle are 4 cm and 5 cm. Then draw a circumcircle of that triangle.

17. Draw an equilateral triangle with side length 6 cm and then draw its incircle. (Only construction marks are required.)

18. Draw an equilateral triangle with each side measuring 7 cm. Then, draw the incircle (inscribed circle) of that triangle.

19. Draw a triangle with sides of 6 cm, 8 cm, and 10 cm. Then draw the incircle of that triangle.

20. In triangle ABC, \(\angle ABC = 90^\circ\), and AB = 6 cm, BC = 8 cm. Find the circumradius of triangle ABC.

21. Draw a right-angled triangle with the two sides adjacent to the right angle measuring 4 cm each. Then draw the circumcircle of the triangle. (Only construction marks are required.)

22. Draw triangle ABC such that BC = 7 cm, AB = 5 cm, and AC = 6 cm. Then draw the circumcircle of triangle ABC. (Only construction marks are required.)

23. Draw a triangle ABC where AB = 6 cm, BC = 10 cm, and \(\angle ABC = 45^\circ\); then draw the circumcircle of the triangle.

24. Draw an equilateral triangle with side length 6 cm and construct its circumcircle. (Only construction markings are required.)

25. Draw the circumcircle of a triangle. (Only the construction diagram needs to be provided.) Here are some diagrams that show how to construct the circumcircle of a triangle using compass and straightedge techniques:

26. Draw the incircle of an equilateral triangle with side length 10 cm. (Only construction marks are required.)

27. 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.)

28. 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.)

29. Draw an isosceles triangle whose base is 5.6 cm and each of the equal sides is 9 cm long. Now, construct the circumcircle of this triangle.