Question

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 1

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=- (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= (y_2-y_1)/(x_2-x_1) \\ \\ Then: \\ m_(AB) = (1-4)/(7-3) \\ m_(AB) = (-3)/(4) \\ m_(AB)=- (3)/(4)`

So its perpendicular line has the following slope:
`m_(BC)= (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= (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= (4)/(3)*(x-3) \\ \\ Replacing\,\,\,\,point\,\,\,\,C=(8,K): \\ K-4=(4)/(3)*(8-3) \\ K-4=(4)/(3)*5 \\ K= (20)/(3) +4 \\ K= (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