Answer:
The values are:
Step-by-step explanation:
Given:
- P = (x₁, y₁, z₁) = (1, 2, b)
- Q = (x₂, y₂, z₂) = (c, -7, 4)
- m = R = (x, y, z) = (-3, a, -1)
To Determine:
a = ?
b = ?
c = ?
Determining the values of a, b, and c
Using the mid-point formula
![m\:=\:\left(\frac{x_1+x_2}{2},\:\frac{y_1+y_2}{2},\:\frac{z_1+z_2}{2}\right)](https://tex.z-dn.net/?f=m%5C%3A%3D%5C%3A%5Cleft%28%5Cfrac%7Bx_1%2Bx_2%7D%7B2%7D%2C%5C%3A%5Cfrac%7By_1%2By_2%7D%7B2%7D%2C%5C%3A%5Cfrac%7Bz_1%2Bz_2%7D%7B2%7D%5Cright%29)
- As the point R(-3, a, -1) is the midpoint of the line segment jointing the points P(1,2,b) and Q(c,-7,4), so
- m = R = (x, y, z) = (-3, a, -1)
Using the mid-point formula
![m\:=\:\left(\frac{x_1+x_2}{2},\:\frac{y_1+y_2}{2},\:\frac{z_1+z_2}{2}\right)](https://tex.z-dn.net/?f=m%5C%3A%3D%5C%3A%5Cleft%28%5Cfrac%7Bx_1%2Bx_2%7D%7B2%7D%2C%5C%3A%5Cfrac%7By_1%2By_2%7D%7B2%7D%2C%5C%3A%5Cfrac%7Bz_1%2Bz_2%7D%7B2%7D%5Cright%29)
given
(x₁, y₁, z₁) = (1, 2, b) = P
(x₂, y₂, z₂) = (c, -7, 4) = Q
m = (x, y, z) = (-3, a, -1) = R
substituting the value of (x₁, y₁, z₁) = (1, 2, b) = P, (x₂, y₂, z₂) = (c, -7, 4) = Q, and m = (x, y, z) = (-3, a, -1) = R in the mid-point formula
![m\:=\:\left(\frac{x_1+x_2}{2},\:\frac{y_1+y_2}{2},\:\frac{z_1+z_2}{2}\right)](https://tex.z-dn.net/?f=m%5C%3A%3D%5C%3A%5Cleft%28%5Cfrac%7Bx_1%2Bx_2%7D%7B2%7D%2C%5C%3A%5Cfrac%7By_1%2By_2%7D%7B2%7D%2C%5C%3A%5Cfrac%7Bz_1%2Bz_2%7D%7B2%7D%5Cright%29)
![\left(x,\:y,\:z\right)\:=\:\left(\frac{1+c}{2},\:\frac{2+\left(-7\right)}{2},\:\frac{b+4}{2}\right)](https://tex.z-dn.net/?f=%5Cleft%28x%2C%5C%3Ay%2C%5C%3Az%5Cright%29%5C%3A%3D%5C%3A%5Cleft%28%5Cfrac%7B1%2Bc%7D%7B2%7D%2C%5C%3A%5Cfrac%7B2%2B%5Cleft%28-7%5Cright%29%7D%7B2%7D%2C%5C%3A%5Cfrac%7Bb%2B4%7D%7B2%7D%5Cright%29)
as (x, y, z) = (-3, a, -1), so
![\left(-3,\:a,\:-1\right)\:=\:\left(\frac{1+c}{2},\:\frac{2+\left(-7\right)}{2},\:\frac{b+4}{2}\right)](https://tex.z-dn.net/?f=%5Cleft%28-3%2C%5C%3Aa%2C%5C%3A-1%5Cright%29%5C%3A%3D%5C%3A%5Cleft%28%5Cfrac%7B1%2Bc%7D%7B2%7D%2C%5C%3A%5Cfrac%7B2%2B%5Cleft%28-7%5Cright%29%7D%7B2%7D%2C%5C%3A%5Cfrac%7Bb%2B4%7D%7B2%7D%5Cright%29)
<u>Determining 'c'</u>
-3 = (1+c) / (2)
-3 × 2 = 1+c
![1+c = -6](https://tex.z-dn.net/?f=1%2Bc%20%3D%20-6)
![c = -6 - 1](https://tex.z-dn.net/?f=c%20%3D%20-6%20-%201)
![c = -7](https://tex.z-dn.net/?f=c%20%3D%20-7)
<u>Determining 'a'</u>
a = (2+(-7)) / 2
![2a = 2-7](https://tex.z-dn.net/?f=2a%20%3D%202-7)
![2a = -5](https://tex.z-dn.net/?f=2a%20%3D%20-5)
![a = -5/2](https://tex.z-dn.net/?f=a%20%3D%20-5%2F2)
<u>Determining 'b'</u>
-1 = (b+4) / 2
![-2 = b+4](https://tex.z-dn.net/?f=-2%20%3D%20b%2B4)
![b = -2-4](https://tex.z-dn.net/?f=b%20%3D%20-2-4)
![b = -6](https://tex.z-dn.net/?f=b%20%3D%20-6)
Therefore, the values are: