Question

Show that the mid-point of line segment joining the points (0, 5) and (15, 7) is same as the mid-point of line segment joining the points (8, −15) and (7, 27).

Answer 1

Midpoint formula:

M between (x₁, y₁) and (x₂, y₂) is M = `((x_1 + x_2) / 2, (y_1 + y_2) / 2)`

⇒ For the points (0, 5) and (15, 7):

Let, x₁ = 0, y₁ = 5;

x₂ = 15, y₂ = 7

Compute x-coordinate:

xM = `(x_1 + x_2) / 2`

= `(0 + 15) / 2`

= `15 / 2`

∴ xM = 7.5

Compute y-coordinate:

yM = `(y_1 + y_2) / 2`

= `(5 + 7) / 2`

= `12 / 2`

∴ yM = 6

Hence, midpoint M1 = `(15/2, 6)` = (7.5, 6)

⇒ For the points (8, −15) and (7, 27):

Let, x₁ = 8, y₁ = −15;

x₂ = 7, y₂ = 27

Compute x-coordinate:

xM = `(x_1 + x_2) / 2`

= `(8 + 7) / 2`

= `15 / 2`

∴ xM = 7.5

Compute y-coordinate:

yM = `(y_1 + y_2) / 2`

= `(−15 + 27) / 2`

= `12 / 2`

∴ yM = 6

Hence, midpoint M2 = `(15/2, 6)` = (7.5, 6)

M1 = M2 = `(15/2, 6)`

Therefore, the midpoint of the segment joining (0,5) and (15,7) is the same as the midpoint of the segment joining (8, −15) and (7, 27).