Problem Analysis:
- We are given an equilateral triangle ABC.
- O is the circumcentre of this triangle.
- The side length is 6 cm.
- We need to find the measure of the angle $\angle BOC$.
Key Geometric Principle:
The angle subtended by an arc at the centre of a circle is double the angle subtended by the same arc at any point on the circumference. In this case, arc BC subtends $\angle BOC$ at the centre O and $\angle BAC$ at the circumference (point A).
Applying the Principle:
In an equilateral triangle, each interior angle measures $60^\circ$. Therefore:
$ \angle BAC = 60^\circ $
Using the angle theorem:
$ \angle BOC = 2 \times \angle BAC $
Calculation:
- Substitute the value of $\angle BAC$: $ \angle BOC = 2 \times 60^\circ $
- Calculate the final angle: $ \angle BOC = 120^\circ $
Note: The side length of 6 cm is not required to determine the angle $\angle BOC$.
Conclusion:
The angle $\angle BOC$ is $120^\circ$. This corresponds to Option D.
Similar Questions
- The adjacent sides of a parallelogram are $4a$ and $3a$. If the angle between them is $60^{\circ}$, then one of the diagonals of the parallelogram will be:
- Every interior angle of a regular octagon is $135^\circ$. Find the exterior angle of the octagon.
- PQ is a diameter of circle whose centre is O. If a point R lies on a circle and $\angle RPO$ is $42^\circ$, then find $\angle RQP$.
- If every interior angle of a regular polygon is $144^\circ$, then the polygon has ______ sides.
- The ratio of the numbers of sides of two regular polygons is $1 : 2$. If each interior angle of the first polygon is $140^\circ$, then the measure of each interior angle of the second polygon is:
- If the sum of the interior angles of a regular polygon is equal to four times the sum of its exterior angles, then what is the number of diagonals in the polygon?
- Find the ratio of the measure of an angle of a regular pentagon to that of a regular octagon.
- What is the measure of each exterior angle of a regular octagon?
- The difference between the interior and exterior angles at a vertex of a regular polygon is $140^\circ$. The number of sides of the polygon is:
- The area of an isosceles right angle triangle is $81 \text{ cm}^2$ . Find the length of its hypotenuse.
Important Questions from Geometry
ABCDEF is a regular hexagon. Side of the hexagon is 36 cm. What is the area of the triangle AOB ?
If ∆ABC ~ ∆DEF, and BC = 4 cm, EF = 5 cm and the area of triangle ABC = 80 cm 2, then the area of the triangle DEF is:
If Δ ABC is right angled at B, AB = 12 cm and ∠CAB = 60°, determine the length of BC.
If ΔABC and ΔDEF are congruent triangles, then which of the following is FALSE?
D and E are points on the sides AB and AC, respectively, of ΔABC such that DE is parallel to BC and AD ∶ DB = 7 ∶ 9. If CD and BE intersect each other at F. then find the ratio of areas of ΔDEF and ΔCBF.
