I have this question on the equation of a line,

ABCD is a RECTANGLE

A = (7,1) B= (3,4) C= (8, K)

What is the value of K?
Instead of telling me the answer could you please tell me how I would go about answering this question and give me a simple "step-by-step" so I can do it again and explain to my friend from school how to do it

(PS: gradient AB= 5)



Answer :

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: [tex]m=- \frac{1}{a} [/tex]

So, the firs thing we have to do is to find the slope of the line AB, the formula of the slope is:  [tex]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} [/tex]

So its perpendicular line has the following slope: [tex]m_{BC}= \frac{4}{3}[/tex]

Now, the general formula of the line is: [tex]y-y_1=m*(x-x_1)[/tex]

We apply it, with the point B and the slope: mBC, so:

[tex]y-y_1=m*(x-x_1) \\ y-4= \frac{4}{3}*(x-3) [/tex]

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:

[tex]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} [/tex]

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