Rectangle and Equations in a coordinate plane

[thunc14]

Problem: Suppose you are making a rectangle with the following constraints:
1. One line segment is described by the equation 3x+2y=6
2. One of the vertices must be (-8,2). This gives the equation of two more lines
3. To get the last equation, consider the perimeter of the rectangle is 10√13 units.

If you are tasked with finding the equations of all four lines, what does not need to be used?
a. Distance Formula
b. Substitution
c. Slope-intercept form
d. Slope formula
e. All of the above are used

I went ahead and tried to solve all 4 equations. The first one is given, and the second and third can be found by using the reciprocal slope and using the points given.
L1: y=(-3/2)x+3
L2: y=(2/3)x+(22/3)
L3: y=-(3/2)x-10

Using the distance formula, you know that distance between the points (-8, 2) and (-2, 6) is 2√13, so those parallel sides of the rectangle are 4√13 of the total 10√13, so the other two sides must be 3√13 each. I know the slope of the 4th line will be (2/3)x, but I am unclear of how to solve the equation so that it fits the parameters.

As for the answer, I'm also a bit confused. In my way I substituted values into the slope intercept form to get the new equations, but I didn't solve a system of equations using substitution, not sure if that's what the problem means.

So my question is, how do I find the equation for the 4th line, and what would be the correct answer? I want to answer D, slope formula just because I didn't count a certain number of units up or down, then a certain number of units left of right, but I'm not sure if that's what they mean by the slope formula. Kind of a weird questions imho.

 

[Dr.Peterson]

I don't think you've told us yet where these "what isn't needed" problems come from. Are they from a textbook that would have taught specific methods they are assuming you would follow, or something more general? The assumptions they seem to make are less justifiable in the latter case. I've already expressed my opinion of this type of problem, regardless of context.

In any case, good work so far. You have used the distance formula (more or less necessary), slope-intercept form (to find the slope, though there are other ways), and maybe substitution (into a formula), depending on what they mean. I'm inclined to agree that you haven't used the "slope formula" (definition of slope).

Now, how to finish? I can think of several approaches from here. There are direct methods to find a parallel line at a given distance from a given line, but they probably assume you don't know those (and I'd have to think a bit, or look them up). You could also use vectors, which I suppose are assumed not to be available to you. So perhaps the most direct way is to suppose a point (x,y) is on L3 at a distance of 3√13 from (-8,2), writing an equation for that distance and solving.

 

[hoosie]

 

[hoosie]