1. The circumcenter of a------ triangle lies outside the triangle.

2. The circumcenter of an acute-angled triangle lies inside the triangle.

(a) inside (b) outside (c) on the hypotenuse (d) on the hypotenuse

3. Given: In triangle △ABC, O is the circumcenter and OD ⊥ BC. Prove that: ∠BOD = ∠BAC Let’s break it down in English: **Given:** In triangle △ABC, O is the circumcenter (the point where the perpendicular bisectors of the sides meet), and OD is perpendicular to side BC. **To Prove:** The angle ∠BOD formed at the center between points B and D is equal to the angle ∠BAC at the vertex A. This is a classic geometry result based on the properties of a circle and triangle. Would you like me to walk you through the full proof in English as well?

4. In the isosceles triangle ABC, AB = AC. The circumcenter of the triangle is O, and it lies on the side of BC opposite to the vertex A. If \(\angle\)BOC = 100°, calculate and write the measures of \(\angle\)ABC and \(\angle\)ABO.

5. If \(O\) is the circumcenter of the equilateral triangle \(ABC\), the value of \(\angle BOC\) will be -

(a) 100° (b) 120° (c) 140° (d) 60°

6. ABC is a triangle. Find the value of \(\sin\left(\cfrac{B+C}{2}\right)\)

(a) sin⁡\(\cfrac{A}{2}\) (b) cos⁡\(\cfrac{A}{2}\) (c) sinA (d) cosA

7. 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

8. In triangle ABC, the circumcenter is O; points A and B, C lie on opposite sides of the center. If \(\angle BOC = 120^\circ\), then what is the measure of \(\angle BAC\)?

(a) 50° (b) 60° (c) 70° (d) 80°

9. 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) \(25^o\)

10. In triangle ABC, the circumcenter is O. If \(\angle\)BAC = 85° and \(\angle\)BCA = 75°, then what is the measure of \(\angle\)OAC?

(a) 70° (b) 40° (c) 110° (d) 140°

11. 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

12. In triangle ∆ABC, O is the circumcenter and ∠BAC = 50°. Find the measure of ∠OBC.

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

14. Prove that if a perpendicular is drawn from the right angle vertex of a right-angled triangle to the hypotenuse, then the two adjacent triangles formed are similar to each other and each is also similar to the original triangle.

15. 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.)

16. Given: AC is the diameter of a circle centered at O, ABC is an inscribed triangle, and OP ⊥ AB (where P lies on the circle). **Prove that:** \[ OP : BC = 1 : 2 \]

17. 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.)

18. POR is a triangle. A line is drawn parallel to side QR through point X, the midpoint of PQ, and it intersects side PR at point Y. Prove that point Y is the midpoint of PR.

19. In triangle ABC, the circumcenter is O. It is given that \(\angle\)BAC = 85° and \(\angle\)BCA = 55°. Find the value of \(\angle\)OAC.

20. 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.

21. 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.

22. In triangle \(\triangle ABC\), O is the circumcenter. If \(\angle OAB = 35^\circ\), then what is the measure of \(\angle ACB\)?

(a) 45° (b) 50° (c) 55° (d) 60°

23. 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

24. In triangle ABC, point D lies on side AC such that AD : DC = 3 : 2. If the area of triangle ABC is 40 square cm, then the area of triangle BDC is —

(a) 16 square cm (b) 24 square cm (c) 30 square cm (d) 36 square cm

25. O is the circumcenter of triangle ABC; if \(\angle\)BOC = 80°, then the measure of \(\angle\)BAC is —.

(a) 40° (b) 160° (c) 130° (d) 110°

26. 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.)

27. In triangle ABC, O is the circumcenter and D is the midpoint of side BC. If \(\angle\)BAC = 40°, find the measure of \(\angle\)BOD.

28. 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.

29. If O is the circumcenter of triangle ABC and \(\angle\)OAB = 50°, then the measure of \(\angle\)ACB is —.

30. From the vertex A of triangle \( \triangle ABC \), a perpendicular AD is drawn to the side BC. If \[ \frac{BD}{DA} = \frac{DA}{DC} \] then prove that triangle ABC is a right-angled triangle.