\( d \overline { A B } : 4 = \sqrt { ( 1 + k ) ^ { 2 } + ( k - 6 ) ^ { 2 } } : 4 \)

To solve the equation \[ \frac{d \overline{AB}}{4} = \frac{\sqrt{(1 + k)^2 + (k - 6)^2}}{4}, \] we can start by simplifying both sides. Since both sides are divided by 4, we can multiply through by 4 to eliminate the denominator: \[ d \overline{AB} = \sqrt{(1 + k)^2 + (k - 6)^2}. \] Next, we can square both sides to remove the square root: \[ (d \overline{AB})^2 = (1 + k)^2 + (k - 6)^2. \] Now, let's expand the right-hand side: 1. Expand \((1 + k)^2\): \[ (1 + k)^2 = 1 + 2k + k^2. \] 2. Expand \((k - 6)^2\): \[ (k - 6)^2 = k^2 - 12k + 36. \] Now, combine these two expansions: \[ (1 + k)^2 + (k - 6)^2 = (1 + 2k + k^2) + (k^2 - 12k + 36). \] Combine like terms: \[ = 1 + 2k + k^2 + k^2 - 12k + 36 = 2k^2 - 10k + 37. \] Now we have: \[ (d \overline{AB})^2 = 2k^2 - 10k + 37. \] This is the equation relating \(d \overline{AB}\) and \(k\). If you have a specific value for \(d \overline{AB}\), you can substitute it in to find \(k\). If you need further assistance or a specific value, please provide more details!