Answer:
The possible values of
are -2.944 and -9.055, respectively.
Step-by-step explanation:
From statement we know that
. By Analytical Geometry, we use the equation of a line segment, which is an application of the Pythagorean Theorem:
![AB = 2\cdot BC](https://tex.z-dn.net/?f=AB%20%3D%202%5Ccdot%20BC)
(1)
Where:
,
,
- x-Coordinates of points A, B and C.
- y-Coordinates of points A, B and C.
![(x_{B}-x_{A})^{2}+(y_{B}-y_{A})^{2} = 4\cdot (x_{C}-x_{B})^{2}+4\cdot (y_{C}-y_{B})^{2}](https://tex.z-dn.net/?f=%28x_%7BB%7D-x_%7BA%7D%29%5E%7B2%7D%2B%28y_%7BB%7D-y_%7BA%7D%29%5E%7B2%7D%20%3D%204%5Ccdot%20%28x_%7BC%7D-x_%7BB%7D%29%5E%7B2%7D%2B4%5Ccdot%20%28y_%7BC%7D-y_%7BB%7D%29%5E%7B2%7D)
Then, we expand and simplify the expression above:
![x_{B}^{2}-2\cdot x_{A}\cdot x_{B} +x_{A}^{2} +y_{B}^{2}-2\cdot y_{A}\cdot y_{B} + y_{A}^{2} = 4\cdot (x_{C}^{2}-2\cdot x_{C}\cdot x_{B}+x_{B}^{2})+4\cdot (y_{C}^{2}-2\cdot y_{C}\cdot y_{B}+y_{B}^{2})](https://tex.z-dn.net/?f=x_%7BB%7D%5E%7B2%7D-2%5Ccdot%20x_%7BA%7D%5Ccdot%20x_%7BB%7D%20%2Bx_%7BA%7D%5E%7B2%7D%20%2By_%7BB%7D%5E%7B2%7D-2%5Ccdot%20y_%7BA%7D%5Ccdot%20y_%7BB%7D%20%2B%20y_%7BA%7D%5E%7B2%7D%20%3D%204%5Ccdot%20%28x_%7BC%7D%5E%7B2%7D-2%5Ccdot%20x_%7BC%7D%5Ccdot%20x_%7BB%7D%2Bx_%7BB%7D%5E%7B2%7D%29%2B4%5Ccdot%20%28y_%7BC%7D%5E%7B2%7D-2%5Ccdot%20y_%7BC%7D%5Ccdot%20y_%7BB%7D%2By_%7BB%7D%5E%7B2%7D%29)
![x_{B}^{2}-2\cdot x_{A}\cdot x_{B} + x_{A}^{2} +y_{B}^{2}-2\cdot y_{A}\cdot y_{B} + y_{A}^{2} = 4\cdot x_{A}^{2}-8\cdot x_{C}\cdot x_{B}+4\cdot x_{B}^{2}+4\cdot y_{C}^{2}-8\cdot y_{C}\cdot y_{B}+4\cdot y_{B}^{2}](https://tex.z-dn.net/?f=x_%7BB%7D%5E%7B2%7D-2%5Ccdot%20x_%7BA%7D%5Ccdot%20x_%7BB%7D%20%2B%20x_%7BA%7D%5E%7B2%7D%20%2By_%7BB%7D%5E%7B2%7D-2%5Ccdot%20y_%7BA%7D%5Ccdot%20y_%7BB%7D%20%2B%20y_%7BA%7D%5E%7B2%7D%20%3D%204%5Ccdot%20x_%7BA%7D%5E%7B2%7D-8%5Ccdot%20x_%7BC%7D%5Ccdot%20x_%7BB%7D%2B4%5Ccdot%20x_%7BB%7D%5E%7B2%7D%2B4%5Ccdot%20y_%7BC%7D%5E%7B2%7D-8%5Ccdot%20y_%7BC%7D%5Ccdot%20y_%7BB%7D%2B4%5Ccdot%20y_%7BB%7D%5E%7B2%7D)
If we know that
,
,
,
,
and
, then we have the following expression:
![1 -10 +25 +b^{2} -12\cdot b+36 = 100 -8 +4 +36+24\cdot b +4\cdot b^{2}](https://tex.z-dn.net/?f=1%20-10%20%2B25%20%2Bb%5E%7B2%7D%20-12%5Ccdot%20b%2B36%20%20%3D%20100%20-8%20%2B4%20%2B36%2B24%5Ccdot%20b%20%2B4%5Ccdot%20b%5E%7B2%7D)
![b^{2}-12\cdot b +52 = 4\cdot b^{2}+24\cdot b +132](https://tex.z-dn.net/?f=b%5E%7B2%7D-12%5Ccdot%20b%20%2B52%20%3D%204%5Ccdot%20b%5E%7B2%7D%2B24%5Ccdot%20b%20%2B132)
![3\cdot b^{2}+36\cdot b +80 = 0](https://tex.z-dn.net/?f=3%5Ccdot%20b%5E%7B2%7D%2B36%5Ccdot%20b%20%2B80%20%3D%200)
This is a second order polynomial, which means the existence of two possible real solutions. By Quadratic Formula, we have the following y-coordinates for point B:
, ![b_{2} \approx -9.055](https://tex.z-dn.net/?f=b_%7B2%7D%20%5Capprox%20-9.055)
In consequence, the possible values of
are -2.944 and -9.055, respectively.