Given:
- Triangle ABC is right angled at B
- BD is perpendicular to AC
- AD = 4 cm
- CD = 5 cm

To Find:
- BD and AB

Explanation:

Step 1: Understanding the Problem
- We are given a right-angled triangle ABC, where the right angle is at B.
- BD is the perpendicular drawn from B to AC.
- We need to find the lengths of BD and AB.

Step 2: Applying Pythagoras Theorem
- In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
- Using Pythagoras Theorem, we have:
AC² = AB² + BC²
BC² = AC² - AB²

Step 3: Finding the Length of AC
- AC is the hypotenuse of the right-angled triangle ABC.
- AC can be found using the Pythagorean theorem by substituting the given values:
AC² = AD² + CD²
AC² = 4² + 5²
AC² = 16 + 25
AC² = 41
AC = √41

Step 4: Finding the Length of BC
- BC is the side adjacent to the right angle and opposite to angle A.
- BC can be found by substituting the known values into the Pythagorean theorem:
BC² = AC² - AB²
BC² = 41 - AB²

Step 5: Finding the Length of BD
- BD is the perpendicular drawn from B to AC.
- BD divides the triangle ABC into two right-angled triangles, ABD and CBD.
- In triangle ABD, BD is the altitude drawn from the right angle.
- In triangle CBD, BD is the hypotenuse.
- Since AD = 4 cm, the length of BD can be found using the Pythagorean theorem in triangle ABD:
AB² + BD² = AD²
AB² + BD² = 4²
AB² + BD² = 16
BD = √(16 - AB²)

Step 6: Solving for AB and BD
- Substituting the value of BC from step 4 into the equation from step 5, we can solve for AB:
BC² = 41 - AB²
AB² + BC² = 41
AB² + (41 - AB²) = 41
AB² - AB² + 41 = 41
41 = 41

- Since the equation simplifies to 41 = 41, we can conclude that AB can take any value less than √41.

Step 7: Final Answer
- The length of BD is given by BD = √(16 - AB²).
- AB can take any value less than √41.

Conclusion:
- The length of BD is given by BD = √(16 - AB²).
- AB can take any value less than √41.