Question

The point R(-3,a,-1) is the midpoint of the line segment jointing the points P(1,2,b)

Answer 1

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)  

• 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

$m\:=\:\left(\frac{x_1+x_2}{2},\:\frac{y_1+y_2}{2},\:\frac{z_1+z_2}{2}\right)$

• 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

$m\:=\:\left(\frac{x_1+x_2}{2},\:\frac{y_1+y_2}{2},\:\frac{z_1+z_2}{2}\right)$

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

$m\:=\:\left(\frac{x_1+x_2}{2},\:\frac{y_1+y_2}{2},\:\frac{z_1+z_2}{2}\right)$

$\left(x,\:y,\:z\right)\:=\:\left(\frac{1+c}{2},\:\frac{2+\left(-7\right)}{2},\:\frac{b+4}{2}\right)$

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

$\left(-3,\:a,\:-1\right)\:=\:\left(\frac{1+c}{2},\:\frac{2+\left(-7\right)}{2},\:\frac{b+4}{2}\right)$

Determining 'c'

-3 = (1+c) / (2)

-3 × 2 = 1+c

$1+c = -6$

$c = -6 - 1$

$c = -7$

Determining 'a'

a = (2+(-7)) / 2

$2a = 2-7$

$2a = -5$

$a = -5/2$

Determining 'b'

-1 = (b+4) / 2

$-2 = b+4$

$b = -2-4$

$b = -6$

Therefore, the values are:

• $a = -5/2$

• $b = -6$

• $c = -7$