Combining equations of two perpendicular bisectors

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Probability

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I have two equations, the first one is y = 7x + 44/2, and the second one is y = - 1x - 22/2

I want to find the value of "x" by combining these two equations, if the word "combining" is chosen correctly?

So I have tried it two ways, they are;

7x + 44/2 = -1x - 22/2

7x + 44 = -1x - 44/2

7x + 88 = -1x - 44

7x = - 1x - 132

x = - 16.5

That was my first way of doing it, then I tried the opposite way by changing the equations round.

-1x - 22/2 = 7x + 44/2

-1x - 22 = 7x + 88/2

-1x - 44 = 7x + 88

-1x = 7x + 132

x = -16.5

I seem to get the same answer each way although I think I am solving for "x" correctly.

If I put in my "x" compoent value into either of the equations I should find the "y" value.

-1(-16.5) - 11 = -16.5 - 11 = -27.5

So the coordinates for my centre of a circle look like (-16.5, -27.5)

The radius would be;

r^2 = (-16.5 - 0)^2 + (-27.5 - 0)^2 = 1028.5

Sqrt = 32

Given originally that my coordinates were A(1, 4) B(-6, 5) and C(-7, 4), where I was asked to find the midpoints of the lines AB and BC, thus the perpendicular bisectors of AB and BC, then find the centre of the circle, I feel that I am not getting something quite right when I get a radius of 32 based on the coordinates above?

I would appreciate any second opinions thanks
 
Probability said:
I have two equations, the first one is y = 7x + 44/2, and the second one is y = - 1x - 22/2

I want to find the value of "x" by combining these two equations, if the word "combining" is chosen correctly?

So I have tried it two ways, they are;

7x + 44/2 = -1x - 22/2

7x + 44 = -1x - 44/2 <-- No.
Click to expand...


You must multiply everything on both sides of the equation by 2 -- not just two of the terms.

2(7x + 44/2) = 2(-1x - 22/2 )

14x + 44 = -2x - 22

Of course, there are many ways to proceed. You could have simplified the fractions first and had

7x + 22 = -1x - 11

Hope that helps.
 
wjm11 said:
You must multiply everything on both sides of the equation by 2 -- not just two of the terms.

2(7x + 44/2) = 2(-1x - 22/2 )

14x + 44 = -2x - 22

Of course, there are many ways to proceed. You could have simplified the fractions first and had

7x + 22 = -1x - 11

Hope that helps.
Click to expand...

Thank you for pointing this error out to me, very much appreciated. I have done these equations that many different ways now that my mind is blind to what is actually right, but your solutions are right and in my previous written rough workings I have had those figures.

When I get the value of "x" and put the value back into the equations I am finding the "y" value I think according to the equations, however on my sketch drawing of the graph these "y" values appear to be too far down the graph, and from the above the "x" value of - 8.25 is or appears to be too far to the left according to the graph, so I must be getting something wrong somewhere?
 
Probability said:
I have two equations, the first one is y = 7x + 44/2, and the second one is y = - 1x - 22/2 <--- This equation is wrong

...
Given originally that my coordinates were A(1, 4) B(-6, 5) and C(-7, 4), where I was asked to find the midpoints of the lines AB and BC, thus the perpendicular bisectors of AB and BC, then find the centre of the circle, I feel that I am not getting something quite right when I get a radius of 32 based on the coordinates above?

I would appreciate any second opinions thanks
Click to expand...

1. Draw a sketch!

2. Determine the midpoints of AB: \(\displaystyle \displaystyle{M_{AB}\left(-\frac52 , \frac92 \right)}\)
and
BC: \(\displaystyle \displaystyle{M_{BC}\left(-\frac{13}2 , \frac92 \right)}\)

3. Determeine the slope of AB and BC and consequently the slope of the perpendicular line:

\(\displaystyle \displaystyle{m_{AB} = \frac{5-4}{-6-1} = -\frac17~\implies~m_{\perp AB} = 7}\)

\(\displaystyle \displaystyle{m_{BC} = \frac{4-5}{-7-(-6)} = 1~\implies~m_{\perp AB} = -1}\)

4. Use the coordinates of the midpoints and the slope of the perpendicular lines to determien the equations of the perpendicular bisectors b:

\(\displaystyle \displaystyle{b_{AB}: y-\frac92=7\left(x-\left(-\frac52 \right) \right)~\implies~y = 7x+22}\)

\(\displaystyle \displaystyle{b_{BC}: y-\frac92=(-1) \left(x-\left(-\frac{13}2 \right) \right)~\implies~y = -x-2}\)

5. Determine the coordinates of the point of intersection of the 2 bisectors:

Solve for x: ........... \(\displaystyle \displaystyle{7x+22 = -x - 2}\)

Plug in the x-value in one ot the 2 equations of the bisectors to get the y-coordinate of the center of circumcircle of the triangle ABC.
 

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Many thanks for all your help very much appreciated. The graph software used is excellent I wouldn't mind knowing what it is?

Thanks:grin:
 
pappus said:
1. Draw a sketch!

2. Determine the midpoints of AB: \(\displaystyle \displaystyle{M_{AB}\left(-\frac52 , \frac92 \right)}\)
and
BC: \(\displaystyle \displaystyle{M_{BC}\left(-\frac{13}2 , \frac92 \right)}\)

3. Determeine the slope of AB and BC and consequently the slope of the perpendicular line:

\(\displaystyle \displaystyle{m_{AB} = \frac{5-4}{-6-1} = -\frac17~\implies~m_{\perp AB} = 7}\)

\(\displaystyle \displaystyle{m_{BC} = \frac{4-5}{-7-(-6)} = 1~\implies~m_{\perp AB} = -1}\)

4. Use the coordinates of the midpoints and the slope of the perpendicular lines to determien the equations of the perpendicular bisectors b:

\(\displaystyle \displaystyle{b_{AB}: y-\frac92=7\left(x-\left(-\frac52 \right) \right)~\implies~y = 7x+22}\)

\(\displaystyle \displaystyle{b_{BC}: y-\frac92=(-1) \left(x-\left(-\frac{13}2 \right) \right)~\implies~y = -x-2}\)

5. Determine the coordinates of the point of intersection of the 2 bisectors:

Solve for x: ........... \(\displaystyle \displaystyle{7x+22 = -x - 2}\)

Plug in the x-value in one ot the 2 equations of the bisectors to get the y-coordinate of the center of circumcircle of the triangle ABC.
Click to expand...

Thanks for your help, but I am struggling to follow this last part;
\(\displaystyle \displaystyle{b_{BC}: y-\frac92=(-1) \left(x-\left(-\frac{13}2 \right) \right)~\implies~y = -x-2}\)

5. Determine the coordinates of the point of intersection of the 2 bisectors:

Solve for x: ........... \(\displaystyle \displaystyle{7x+22 = -x - 2}\)

I get -1x + 11, but I can't follow yours through?
 
Probability said:
Thanks for your help, but I am struggling to follow this last part;
\(\displaystyle \displaystyle{b_{BC}: y-\frac92=(-1) \left(x-\left(-\frac{13}2 \right) \right)~\implies~y = -x-2}\)
Click to expand...

Expand the bracket:

\(\displaystyle \displaystyle{b_{BC}: y-\frac92=-x-\frac{13}2 ~\implies~y = -x-\frac{13}2 + \frac92}~\implies~\boxed{y=-x-2}\)


5. Determine the coordinates of the point of intersection of the 2 bisectors:

Solve for x: ........... \(\displaystyle \displaystyle{7x+22 = -x - 2}\)

I get -1x + 11, but I can't follow yours through?
Click to expand...

At the point of intersection the y-values of two different equations must be equal, thus

\(\displaystyle \displaystyle{7x+22 = -x - 2}\)

Now add x at both sides and subtract 22 at both sides of the equation. You'll get:

\(\displaystyle \displaystyle{8x = - 24 ~\implies~ \boxed{x=-3}}\)

Consequently the y-value is \(\displaystyle \displaystyle{y = -(-3)-2 ~\implies~ \boxed{y=1} }\)

Compare with the result you can get by my sketch.
 
Probability said:
Many thanks for all your help very much appreciated. The graph software used is excellent I wouldn't mind knowing what it is?

Thanks:grin:
Click to expand...

Have a look here: http://www.dynageo.de/index.html

This program is used in German schools. The "English" version is poorly translated and hardly usable without any training.

If I were you I would study these programs: http://en.wikipedia.org/wiki/List_of_interactive_geometry_software

and take the one you can master without difficulties.
 
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