26. Let $$ \mathrm{O}(0,0), \mathrm{P}(3,4), \mathrm{Q}(6,0) $$ be the vertices of the triangle OPQ . The point R inside the triangle OPQ is such that the triangles $$ \mathrm{OPR}, \mathrm{PQR}, \mathrm{OQR} $$ are of equal area. The co-ordinates of R is $$ \left(k, \frac{4}{5}\right) $$, then $$ \mathrm{k}= $$

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To find the value of $$ k $$ such that the point $$ R(k, \frac{4}{5}) $$ divides the triangle $$ OPQ $$ into three smaller triangles $$ OPR $$, $$ PQR $$, and $$ OQR $$ of equal area, we can follow these steps:

1. **Calculate the area of triangle $$ OPQ $$**:
   The vertices of triangle $$ OPQ $$ are $$ O(0,0) $$, $$ P(3,4) $$, and $$ Q(6,0) $$. The area $$ A $$ of triangle $$ OPQ $$ can be calculated using the formula:
   $$
   A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
   $$
   where $$ (x_1, y_1) = (0, 0) $$, $$ (x_2, y_2) = (3, 4) $$, and $$ (x_3, y_3) = (6, 0) $$.

2. **Substituting the coordinates into the area formula**:
   $$
   A = \frac{1}{2} \left| 0(4 - 0) + 3(0 - 0) + 6(0 - 4) \right|
   $$
   $$
   = \frac{1}{2} \left| 0 + 0 - 24 \right| = \frac{1}{2} \times 24 = 12
   $$

3. **Area of each smaller triangle**:
   Since the area of triangle $$ OPQ $$ is 12, the area of each of the smaller triangles $$ OPR $$, $$ PQR $$, and $$ OQR $$ must be:
   $$
   \text{Area of each triangle} = \frac{12}{3} = 4
   $$

4. **Calculate the area of triangle $$ OPR $$**:
   The coordinates of $$ R $$ are $$ (k, \frac{4}{5}) $$. The area of triangle $$ OPR $$ can be calculated as:
   $$
   A_{OPR} = \frac{1}{2} \left| 0(4 - \frac{4}{5}) + 3(\frac{4}{5} - 0) + k(0 - 4) \right|
   $$
   $$
   = \frac{1}{2} \left| 0 + \frac{12}{5} - 4k \right| = \frac{1}{2} \left| \frac{12}{5} - 4k \right|
   $$

5. **Setting the area of triangle $$ OPR $$ equal to 4**:
   $$
   \frac{1}{2} \left| \frac{12}{5} - 4k \right| = 4
   $$
   $$
   \left| \frac{12}{5} - 4k \right| = 8
   $$

6. **Solving the absolute value equation**:
   This gives us two cases:
   - Case 1: $$ \frac{12}{5} - 4k = 8 $$
   - Case 2: $$ \frac{12}{5} - 4k = -8 $$

**Case 1**:
   $$
   \frac{12}{5} - 4k = 8
   $$
   $$
   -4k = 8 - \frac{12}{5}
   $$
   $$
   -4k = \frac{40}{5} - \frac{12}{5} = \frac{28}{5}
   $$
   $$
   k = -\frac{7}{5}
   $$

**Case 2**:
   $$
   \frac{12}{5} - 4k = -8
   $$
   $$
   -4k = -8 - \frac{12}{5}
   $$
   $$
   -4k = -\frac{40}{5} - \frac{12}{5} = -\frac{52}{5}
   $$
   $$
   k = \frac{13}{5}
   $$

7. **Conclusion**:
   The possible values for $$ k $$ are $$ -\frac{7}{5} $$ and $$ \frac{13}{5} $$. Since $$ R $$ must be inside triangle $$ OPQ $$, we need to check which of these values is valid.

The point $$ R(k, \frac{4}{5}) $$ must satisfy the conditions of being inside the triangle formed by $$ O(0,0) $$, $$ P(3,4) $$, and $$ Q(6,0) $$. The valid value for $$ k $$ that keeps $$ R $$ inside the triangle is $$ k = \frac{13}{5} $$.

Thus, the value of $$ k $$ is:
$$
\boxed{\frac{13}{5}}
$$
$$ k = \frac{13}{5} $$

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