The perimeter of a triangle ABC is 27 cm. The length of side BC is 12 cm. If its vertex A is connected with its incentre I and extended towards BC so that the connecting line intersects BC at D, then the ratio of AI : DI is equal to

The problem is related to the properties of the incenter of a triangle. The incenter of a triangle is the center of its inscribed circle (incircle). It is the point where the angle bisectors of the triangle intersect.

The incenter is equally distant from the three sides of the triangle. This distance is called the inradius (r).

The incenter divides the line segment from the vertex to the point where the inradius touches the opposite side in the ratio of the lengths of the two sides forming the angle at the vertex.

Let's denote the sides of the triangle as follows: AB = a, BC = b, and AC = c.

Given that BC = 12 cm and the perimeter of the triangle is 27 cm, we can find the lengths of the other sides. The perimeter of a triangle is the sum of the lengths of its sides, so we have a + b + c = 27 cm. Substituting b = 12 cm, we get a + c = 27 - 12 = 15 cm.

We don't know the exact lengths of a and c, but we know that the ratio of AI to DI is the same as the ratio of a to c.

So, the ratio of AI to DI is equal to a : c, which is the same as the ratio of the lengths of the sides AB to AC.

Without the exact lengths of a and c, we can't find a specific numerical ratio, but we know that the ratio is the same as the ratio of the lengths of the sides AB to AC.