To solve this problem, let's first clarify what is given:

• You have a triangle $ABC$ with $AB = 5$ cm and $AC = 15$ cm.
• Point $D$ is on side $BC$, and $DA$ bisects the triangle $ABC$ into two parts: $\triangle ABD$ and $\triangle ACD$.

You are asked to find the ratio of the areas of $\triangle ABC$ and $\triangle ACD$.

Solution

1. Understanding the bisector: Since $DA$ is a bisector, the area of $\triangle ABD$ is equal to the area of $\triangle ACD$ if $D$ were the midpoint of $BC$. But in this problem, it is given that $DA$ bisects $\triangle ABC$.

2. Ratio of areas:

   • The area of $\triangle ABC$ is proportional to the product of the base and the height. If we consider $\triangle ACD$, its base is $CD$ and its height is the same as $\triangle ABC$ because $DA$ is a bisector.

   • The ratio of the areas of two triangles with the same height is equal to the ratio of their bases.

3. Use the Side Lengths:

   • Let's consider the side lengths $AB$ and $AC$ to find the ratio of the areas. Since $DA$ is a bisector, it divides the opposite side in the ratio of the adjacent sides.
   • Thus, the ratio $BD:DC$ will be equal to $AB:AC = 5:15 = 1:3$.

4. Area Ratio:

   • The area of $\triangle ACD$ is smaller because $DC$ is smaller than $BC$.
   • Since the area ratio is given by the square of the corresponding side ratio, the area ratio of $\triangle ABC$ to $\triangle ACD$ is $1:3$.

Conclusion

The ratio of the areas of $\triangle ABC$ and $\triangle ACD$ is $3:2$.

Would you like details or have any questions?

Here are five related questions:

1. What is the significance of the angle bisector theorem in triangle area calculations?
2. How do you find the area of a triangle using side lengths?
3. What is the relationship between side lengths and area ratios in similar triangles?
4. How can you apply the angle bisector theorem in different types of triangles?
5. How does the area ratio change if $D$ is not on $BC$ but on an extended line?

Tip: When solving geometry problems, always consider theorems like the angle bisector theorem and properties of triangles to guide you through the solution.