how to do this without cheating?

[allegansveritatem]

Problem (only part C is relevant for this post):


I knew from working out A and B that the distance between R and Q was sqrt of 265. But I couldn't figure out how to find the coordinates of R without knowing at least one of them and didn't know how to get that info from what had been given or what had been derived (distance between P and Q, and midpoint) but I tried to do it the hard way because I think that is how the author wanted it done. Here is what I did:
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I did not have this worked out in the solutions book--it is an even numbered problem--but I found the answer in the back of the text because it is part of a review test. Here is the solution:


So I got my answer but I didn't get it without fudging. If I had to solve a problem like this without a graph, how would I do it? I have a feeling it may havve something to do with using points P and Q to get the slope of the line joining them and use the negative reciprocal of that slope and the coordinates of Q to find the equation of the perpendicular bisector to line between P and Q and then maybe....what? Or is that too baroque altogether?

 

[pka]

Problem (only part C is relevant for this post):
View attachment 12258

If \(\displaystyle R: (a,b)\) then the mid-point of \(\displaystyle \overline{PR}\) is \(\displaystyle \left(\frac{-5+a}{2},\frac{9+b}{2}\right)\).
That means that \(\displaystyle \frac{-5+a}{2}=-8~\&~\frac{9+b}{2}=-7\)
Surely you can solve for \(\displaystyle (a,b)\)???

 

[Dr.Peterson]

I assume that this exercise was given after teaching the formula for the midpoint of a segment, namely that the point midway between (a,b) and (c,d) is ((a+c)/2, (b+d)/2). If not, quite likely you are expected to have discovered that in solving part (b).

Given that formula, you can suppose that R is (x,y), and the fact that Q is the midpoint of PR can be written as

((-5 + x)/2, (9 + y)/2) = (-8, -7)​

This is really two equations:

(-5 + x)/2 = -8​

(9 + y)/2 = -7​

Solve those, and you have R.

It can be solved even more easily using vectors, but I assume you have not learned about those.

 

[allegansveritatem]

If \(\displaystyle R: (a,b)\) then the mid-point of \(\displaystyle \overline{PR}\) is \(\displaystyle \left(\frac{-5+a}{2},\frac{9+b}{2}\right)\).
That means that \(\displaystyle \frac{-5+a}{2}=-8~\&~\frac{9+b}{2}=-7\)
Surely you can solve for \(\displaystyle (a,b)\)???

yes, I did the a and b part:

 

[allegansveritatem]

I assume that this exercise was given after teaching the formula for the midpoint of a segment, namely that the point midway between (a,b) and (c,d) is ((a+c)/2, (b+d)/2). If not, quite likely you are expected to have discovered that in solving part (b).

Given that formula, you can suppose that R is (x,y), and the fact that Q is the midpoint of PR can be written as

((-5 + x)/2, (9 + y)/2) = (-8, -7)​

This is really two equations:

(-5 + x)/2 = -8​

(9 + y)/2 = -7​

Solve those, and you have R.

It can be solved even more easily using vectors, but I assume you have not learned about those.

Well I did find the midpoint and the distance between the two given points but I didn't post the resuts in my original post because that might have been too many images. I will post it here--as I already did for the above post:


I have not learned about vectors yet. I want to study your post in the morning because my brain is broken by this time of night and needs to sleep to be repaired. But I can see dimly that you are presenting what I want to know. I will get back to this thread tomorrow.

 

[allegansveritatem]

If \(\displaystyle R: (a,b)\) then the mid-point of \(\displaystyle \overline{PR}\) is \(\displaystyle \left(\frac{-5+a}{2},\frac{9+b}{2}\right)\).
That means that \(\displaystyle \frac{-5+a}{2}=-8~\&~\frac{9+b}{2}=-7\)
Surely you can solve for \(\displaystyle (a,b)\)???

I have to study your post in the morning...too brainfagged to do it now. Thanks for the tip.

 

[allegansveritatem]

I copied both contributions to this thread, pasted them to a notepad document and printed them this morning.When time came to do my daily algebra I got the the document and studied it. It took about a minute to see that both contributors were proposing the same solution and I was stunned that I hadn't thought of it myself! I had the info I needed but the solution had been hidden in plain sight. Anyway, thanks to pka and Dr Peterson, here is solution to C) :