Hello,
We have the rectangle ABCD, we know that all the angles of a rectangle are 90º, that means that its consecutive lines are perpendicular.
We also know that the slope of a perpendicular line is the inverse negative, i'm going to make and example, so you can understand:
If a line have a slope: m=a
Its perpendicular line's slope is: \(m=- \frac{1}{a}\)
So, the firs thing we have to do is to find the slope of the line AB, the formula of the slope is: \(m= \frac{y_2-y_1}{x_2-x_1} \\ \\ Then: \\ m_{AB} = \frac{1-4}{7-3} \\ m_{AB} = \frac{-3}{4} \\ m_{AB}=- \frac{3}{4}\)
So its perpendicular line has the following slope: \(m_{BC}= \frac{4}{3}\)
Now, the general formula of the line is: \(y-y_1=m*(x-x_1)\)
We apply it, with the point B and the slope: mBC, so:
\(y-y_1=m*(x-x_1) \\ y-4= \frac{4}{3}*(x-3)\)
This is the function that represents the line BC, We know that the point C is part of the line, then it must meet the equality, so we replace this point and get K:
\(y-4= \frac{4}{3}*(x-3) \\ \\ Replacing\,\,\,\,point\,\,\,\,C=(8,K): \\ K-4=\frac{4}{3}*(8-3) \\ K-4=\frac{4}{3}*5 \\ K= \frac{20}{3} +4 \\ K= \frac{32}{3}\)
If I just explained it, you wouldn't have understood it, that's why I solved it, I hope you undestand, If you have any question, don't hesitate to tell me.
Answer: K=32/3
