Given:
In the given figure, triangle ABC is right-angled at B. The lengths of the legs are 8 cm and 6 cm.

To Find:
We need to find the length of the perpendicular BN on the side AC.

Solution:
We can solve this problem by using the Pythagorean theorem and basic trigonometry.

Step 1: Identify the Right Angle
In triangle ABC, angle B is a right angle.

Step 2: Apply the Pythagorean Theorem
According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In triangle ABC, let's label the hypotenuse as AC, and the other two sides as AB and BC.

Using the Pythagorean theorem, we have:
AC^2 = AB^2 + BC^2

Substituting the given values, we get:
AC^2 = 8^2 + 6^2
AC^2 = 64 + 36
AC^2 = 100
AC = 10 cm

Step 3: Identify the Right-Angled Triangle
In triangle ABC, we have a right-angled triangle.

Step 4: Identify the Perpendicular and Base
In a right-angled triangle, the side opposite the right angle is called the hypotenuse. The sides adjacent to the right angle are called the perpendicular and base.

In triangle ABC, the perpendicular is BN, and the base is AC.

Step 5: Apply Trigonometry
Since we have identified the perpendicular and base, we can use trigonometry to find the length of BN.

In triangle ABC, we can use the trigonometric function tangent (tan) to find the length of BN.

tan(angle) = perpendicular / base

tan(B) = BN / AC
tan(90°) = BN / 10
Infinity = BN / 10

Since the tangent of 90° is undefined, we can conclude that the length of BN is infinity.

Answer:
The length of perpendicular BN on the side AC is infinity.

Let bn=x