To find the value of Δ2 - c2, we need to calculate the area of the triangle (Δ) and the length of the third side (c).

Given information:
Side a = 4
Side b = 3

Finding the area of the triangle (Δ):
To find the area of the triangle, we can use the formula: Δ = (1/2) * base * height

In triangle ABC, let AD be the median from A to BC, and BE be the median from B to AC. Given that AD and BE are mutually perpendicular, we can consider triangle ABC as a right-angled triangle with AD and BE as the legs.

Let M be the midpoint of BC, so AD = DM and BE = EM. Since the medians divide each other into a 2:1 ratio, we can say that DM = 2AD and EM = 2BE.

Using the Pythagorean theorem, we can find the lengths of AD and BE:
AD² + DM² = AM²
BE² + EM² = BM²

Since DM = 2AD and EM = 2BE, we can write the above equations as:
AD² + (2AD)² = AM²
BE² + (2BE)² = BM²

Simplifying, we get:
5AD² = AM²
5BE² = BM²

Now, let's find the lengths of AM and BM:
AM = AB - BM
BM = AB - AM

Using the median property, we know that BM = 2AM. Substituting this into the equation above, we get:
2AM = AB - AM
3AM = AB
AM = AB/3

Similarly, we can find BM:
BM = 2AM
BM = 2(AB/3)
BM = 2AB/3

Substituting these values into the equations for AM² and BM², we get:
5AD² = (AB/3)²
5BE² = (2AB/3)²

Simplifying, we get:
5AD² = AB²/9
5BE² = 4AB²/9

Since AD and BE are medians, they divide the triangle into six smaller triangles of equal area. Therefore, the area of triangle ABC is given by:
Δ = 6 * (1/2) * AD * BE

Substituting the values of AD and BE, we get:
Δ = 6 * (1/2) * (AB/3) * (2AB/3)
Δ = AB²/3

Since the area of triangle ABC is Δ = AB²/3, we can find the value of AB using the given side lengths:
AB² = a² + b²
AB² = 4² + 3²
AB² = 16 + 9
AB² = 25
AB = 5

Substituting AB = 5 into the equation for the area of triangle ABC, we get:
Δ = 5²/3
Δ = 25/3

Finding the length of the third side (c):
To find the length of the third side, c, we can use the law of cosines:
c² = a² + b² - 2ab * cos(C)

Substituting the given values