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The problem involves triangle \(ABC\), which is right-angled at \(B\), and segment \(BD\) is perpendicular to \(AC\). We are given the dimensions of the triangle:

- \(AB = 12 \, \text{cm}\)
- \(BC = 5 \, \text{cm}\)
- \(BD \perp AC\)

We are asked to find:

1. \(\cos \angle DBC\)
2. \(\cot \angle DBA\)

### Step-by-Step Solution

#### 1. To find \(\cos \angle DBC\):
In right-angled triangle \(ABC\), we can first find the hypotenuse \(AC\) using the Pythagorean theorem:

\[
AC = \sqrt{AB^2 + BC^2} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \, \text{cm}
\]

Now consider triangle \(BDC\), which is a right-angled triangle. We need the lengths of \(BD\) and \(DC\), which we don't have directly. However, since \(BD \perp AC\), we could use similar triangles \(ABC\) and \(BDC\) to determine relationships. Let's compute the cosine based on known dimensions.

We know from the cosine definition in a right triangle:

\[
\cos \angle DBC = \frac{\text{Adjacent side}}{\text{Hypotenuse}} = \frac{BD}{BC}
\]

We'll need to calculate or estimate \(BD\) from other given information or using properties of similar triangles.

#### 2. To find \(\cot \angle DBA\):
In triangle \(ABD\), we use the definition of cotangent:

\[
\cot \theta = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{AB}{BD}
\]

Thus, for \(\cot \angle DBA\), we will again need to calculate \(BD\).

---

Next, I'll determine \(BD\) through geometric relationships and solve for the required values.The hypotenuse \(AC\) of triangle \(ABC\) is \(13 \, \text{cm}\).

Next, we will use the fact that triangle \(ABC\) is similar to triangle \(BDC\), which will help us find the length of \(BD\) and complete the calculations for \(\cos \angle DBC\) and \(\cot \angle DBA\).

Let's proceed with finding the specific values of \(BD\) and \(DC\) using similarity properties.It seems like I can't do more advanced data analysis right now. Please try again later.

Would you like me to explain how you could manually solve for the missing values \(BD\) and \(DC\) using similar triangles or any other step-by-step clarification? 

Here are some related questions for deeper understanding:
1. How do you calculate trigonometric ratios in a right-angled triangle?
2. What are the properties of similar triangles?
3. How do you find the length of a perpendicular in a right-angled triangle?
4. How can the Pythagorean theorem be applied to complex problems involving multiple right triangles?
5. What is the relationship between sine, cosine, and cotangent?

**Tip**: In triangle similarity, corresponding angles are equal, and corresponding sides are proportional.

In the diagram, triangle ABC is right-angled at B, and BD is perpendicular to AC. We are given: AB = 12 cm, BC = 5 cm. Find: (i) cos ∠DBC (ii) cot ∠DBA.

Learn how to calculate cos ∠DBC and cot ∠DBA in a right-angled triangle, where BD is perpendicular to AC. This problem involves trigonometric ratios and the Pythagorean theorem, suitable for Grades 9-11.

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