In this question, we can use Heron's formula to find the area of the triangle and then use the formula for the area of a triangle to find the length of RS.

Heron's formula states that the area of a triangle with sides of length a, b, and c is:

\(A = \sqrt{s(s-a)(s-b)(s-c)}\),

where s is the semi perimeter of the triangle, given by \(s = \frac{a+b+c}{2}\).

In this case, we have \(a = PQ = 10\), \(b = QR = 8\), and \(c = RP = 6\), so we can find the semiperimeter as follows:

\(s = \frac{a+b+c}{2} = \frac{10+8+6}{2} = 12\).

Using Heron's formula, we can now find the area of the triangle:

\(A = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{12(12-10)(12-8)(12-6)} = \sqrt{12 \times 2 \times 4 \times 6} = \sqrt{576} = 24\).

The formula for the area of a triangle is \(A = \frac{bh}{2}\), where \(b\) is the length of the base and \(h\) is the height of the triangle (the perpendicular distance from the base to the opposite vertex).

In this case, we can take PQ as the base, so \(b = PQ = 10\). Solving for \(h\),

we get \(h = \frac{2A}{b} = \frac{2 \times 24}{10} = 4.8\).

Since RS is perpendicular to PQ, its length is equal to the height of the triangle, so we have \(RS = 4.8\).