Answer:
The values are:
a = -5/2
b = -6
c = -7
Step-by-step explanation:
Given
P = (x₁, y₁, z₁) = (1, 2, b)
Q = (x₂, y₂, z₂) = (c, -7, 4)
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
[tex]m\:=\:\left(\frac{x_1+x_2}{2},\:\frac{y_1+y_2}{2},\:\frac{z_1+z_2}{2}\right)[/tex]
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 values in the mid-point formula
[tex]m\:=\:\left(\frac{x_1+x_2}{2},\:\frac{y_1+y_2}{2},\:\frac{z_1+z_2}{2}\right)[/tex]
[tex]\left(x,\:y,\:z\right)\:=\:\left(\frac{1+c}{2},\:\frac{2+\left(-7\right)}{2},\:\frac{b+4}{2}\right)[/tex]
as (x, y, z) = (-3, a, -1), so
[tex]\left(-3,\:a,\:-1\right)\:=\:\left(\frac{1+c}{2},\:\frac{2+\left(-7\right)}{2},\:\frac{b+4}{2}\right)[/tex]
so solving 'c'
-3 = (1+c) / (2)
-3 × 2 = 1+c
1+c = -6
c = -6 - 1
c = -7
solving 'a'
a = (2+(-7)) / 2
2a = 2-7
2a = -5
a = -5/2
solving b
-1 = (b+4) / 2
-2 = b+4
b+4 = -2
b = -2-4
b = -6
Thus, the values are:
a = -5/2
b = -6
c = -7
Verification:
[tex]\left(x,\:y,\:z\right)\:=\:\left(\frac{1+c}{2},\:\frac{2+\left(-7\right)}{2},\:\frac{b+4}{2}\right)[/tex]
[tex]\left(-3,\:a,\:-1\right)\:=\:\left(\frac{1+c}{2},\:\frac{2+\left(-7\right)}{2},\:\frac{b+4}{2}\right)[/tex]
put a = -5/2, b = -6, c = -7
[tex]\left(-3,\:-\frac{5}{2},\:-1\right)\:=\:\left(\frac{1+\left(-7\right)}{2},\:-\frac{5}{2},\:\frac{\left(-6\right)+4}{2}\right)[/tex]
[tex]\left(-3,\:-\frac{5}{2},\:-1\right)\:=\:\left(\frac{-6}{2},\:-\frac{5}{2},\:\frac{-2}{2}\right)[/tex]
[tex]\left(-3,\:-\frac{5}{2},\:-1\right)\:=\:\left(-3,\:-\frac{5}{2},\:-1\right)[/tex]
