Let's work through the solution step-by-step based on the given details and geometry principles.

Given Details

1. $CD = DE = 5 \, \text{cm}$
2. $AC = 13 \, \text{cm}$, $AD = 12 \, \text{cm}$
3. Area of $\triangle ABC = 31.2 \, \text{cm}^2$
4. Angle $\angle ADB = 90^\circ$, $BE = 33.8 \, \text{cm}$ (not to use while calculating).

1. Calculating $BE$ (Verification)

Since $CD = DE = 5 \, \text{cm}$, $C$ and $D$ divide $BE$ into three equal parts: $BE = BC + CD + DE$ $BE = 3 \times 5 = 15 \, \text{cm}$ However, the total $BE$ provided is $33.8 \, \text{cm}$, so this discrepancy indicates something might be different in splitting BE into segments.

2. Finding the Angle $ADB$

Given that $\angle ADB = 90^\circ$, we can confirm that $\triangle ADB$ is a right triangle.

By the Pythagorean Theorem for $\triangle ADB$: $AB^2 = AD^2 + BD^2$ Let’s calculate $BD$: $BD = \sqrt{AB^2 - AD^2}$ Substitute $AB = 13 \, \text{cm}$ and $AD = 12 \, \text{cm}$: $BD = \sqrt{13^2 - 12^2} = \sqrt{169 - 144} = \sqrt{25} = 5 \, \text{cm}$

So, $\triangle ADB$ is confirmed to be a right triangle with $\angle ADB = 90^\circ$.

3. Verifying $BE = 33.8 \, \text{cm}$

Since $CD = DE = 5 \, \text{cm}$ and $BE$ is divided proportionally, we need to confirm if these measurements align with $BE = 33.8 \, \text{cm}$.

Using the right triangle relationships and area calculations, further precision could be explored depending on the context.

Would you like clarification on a specific step or additional details on dividing $BE$ into proportional segments?

Follow-up Questions

1. How does the division of $BE$ into segments align with the total given length of $33.8 \, \text{cm}$?
2. What additional relationships between triangles $\triangle ABE$ and $\triangle ABC$ can be explored?
3. Can the proportionality of $BE$ verify the given area $31.2 \, \text{cm}^2$?
4. Would you like detailed confirmation of $\angle ADB = 90^\circ$ with trigonometric reasoning?
5. Should I explore further applications of the given values in trigonometry or geometry?

Tip:

Always cross-check given measurements with derived relationships in geometry problems, especially when proportional segments are involved.