Introduction:
In this problem, we are given a cantilever beam with a square cross-section of side length 'a'. The beam is subjected to an end load, and we are asked to determine the effect of increasing the side length by 19% on the tip deflection.

Given:
- Square cross-section of side length 'a'
- End load applied to the beam
- Side length is increased by 19%
- We need to determine the change in tip deflection

Analysis:
To understand the effect of increasing the side length on the tip deflection, we need to consider the relationship between the deflection and the dimensions of the beam. The deflection of a cantilever beam is influenced by the following factors:

1. Length of the beam: Longer beams tend to have larger deflections.
2. Moment of inertia of the cross-section: Higher moment of inertia leads to smaller deflections.
3. Modulus of elasticity of the material: Higher modulus of elasticity results in smaller deflections.
4. Applied load: Larger loads cause larger deflections.

Relationship between side length and moment of inertia:
The moment of inertia of a square cross-section is given by the formula: I = (a^4)/12, where 'a' is the side length. From this formula, we can see that the moment of inertia is directly proportional to the fourth power of the side length.

Effect of increasing the side length:
When the side length is increased by 19%, the new side length becomes a + 0.19a = 1.19a. Substituting this value in the formula for moment of inertia, we get the new moment of inertia as (1.19a)^4/12 = 1.19^4/12 * a^4.

Change in moment of inertia:
To determine the change in moment of inertia, we can calculate the ratio of the new moment of inertia to the original moment of inertia:
Change in moment of inertia = (1.19^4/12 * a^4) / (a^4/12) = 1.19^4.

Relationship between moment of inertia and deflection:
The deflection of a cantilever beam is inversely proportional to the moment of inertia. Therefore, if the moment of inertia increases, the deflection decreases.

Conclusion:
In this problem, when the side length is increased by 19%, the moment of inertia increases by a factor of 1.19^4. Since the deflection is inversely proportional to the moment of inertia, the deflection decreases by the reciprocal of this factor, which is approximately 1/1.19^4 = 0.5 or 50%. Therefore, the correct answer is option 'D', the tip deflection decreases by 50%.