Question

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)
Find the values of:
a=
b=
C=

Answer 1

The values are:

a = -5/2

b = -6

c = -7

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

`m\:=\:\left((x_1+x_2)/(2),\:(y_1+y_2)/(2),\:(z_1+z_2)/(2)\right)`

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

`m\:=\:\left((x_1+x_2)/(2),\:(y_1+y_2)/(2),\:(z_1+z_2)/(2)\right)`

`\left(x,\:y,\:z\right)\:=\:\left((1+c)/(2),\:(2+\left(-7\right))/(2),\:(b+4)/(2)\right)`

as (x, y, z) = (-3, a, -1), so

`\left(-3,\:a,\:-1\right)\:=\:\left((1+c)/(2),\:(2+\left(-7\right))/(2),\:(b+4)/(2)\right)`

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:

`\left(x,\:y,\:z\right)\:=\:\left((1+c)/(2),\:(2+\left(-7\right))/(2),\:(b+4)/(2)\right)`

`\left(-3,\:a,\:-1\right)\:=\:\left((1+c)/(2),\:(2+\left(-7\right))/(2),\:(b+4)/(2)\right)`

put a = -5/2, b = -6, c = -7

`\left(-3,\:-(5)/(2),\:-1\right)\:=\:\left((1+\left(-7\right))/(2),\:-(5)/(2),\:(\left(-6\right)+4)/(2)\right)`

`\left(-3,\:-(5)/(2),\:-1\right)\:=\:\left((-6)/(2),\:-(5)/(2),\:(-2)/(2)\right)`

`\left(-3,\:-(5)/(2),\:-1\right)\:=\:\left(-3,\:-(5)/(2),\:-1\right)`