the distance of the circumcentre of the acute angled triangle ABC from the sides BC,CA& AB are in the ratio,

1)asinA:bsinB:csinC

2)cosA:cosB:cosC

3)acotA:bcotB:ccotC

4)none of these

To find the correct ratio of the distances from the circumcenter of an acute-angled triangle ABC to its sides, we need to understand a few concepts about circumcenters and distances from points to lines. The circumcenter is the point where the perpendicular bisectors of the sides of the triangle meet, and it is equidistant from all three vertices of the triangle.

Understanding the Distances

The distance from the circumcenter (let's denote it as O) to each side of the triangle can be calculated using specific formulas. For any triangle, the distance (d) from a point to a line can be represented in terms of the area of the triangle and the length of the base (side of the triangle) opposite to the vertex.

Deriving the Distances

Given triangle ABC, the distances from the circumcenter O to the sides BC, CA, and AB can be expressed using the area (Δ) of the triangle and the respective sides:

• Distance to side BC: d_a = (2Δ) / a
• Distance to side CA: d_b = (2Δ) / b
• Distance to side AB: d_c = (2Δ) / c

Where a, b, and c are the lengths of sides BC, CA, and AB, respectively.

Exploring the Ratios

Now, we can analyze the ratios provided in your question:

• asinA : bsinB : csinC - This ratio corresponds to the angles and the sides of the triangle, and it is indeed related to the circumradius and distances from the circumcenter.
• cosA : cosB : cosC - While this ratio involves the angles, it does not specifically relate to the distances from the circumcenter.
• acotA : bcotB : ccotC - This ratio is more complex and does not directly correspond to the distances from the circumcenter.
• none of these - This option suggests that none of the provided ratios are correct.

Which Ratio is Correct?

After analyzing the ratios, we find that the correct answer is the first option: asinA : bsinB : csinC. This ratio is significant because it relates to the areas of the triangles formed by dropping perpendiculars from the circumcenter to the sides. The area of triangle ABC can also be expressed in terms of the circumradius and the sine of the angles, which ties everything together.

Final Thoughts

In summary, the distances from the circumcenter to the sides of an acute-angled triangle are best represented by the ratio of asinA : bsinB : csinC. Understanding these relationships not only helps in solving this question but also deepens your overall comprehension of triangle geometry and trigonometric principles.