Question

Show that the line segment joining the points (1, 5) and (3, −5) is perpendicular to the line segment joining the points (0,3) and (−5, 2).

Answer 1

The slope of a line passing through two points (x₁, y₁) and (x₂, y₂):

`m = (y_2 - y_1)/(x_2 - x_1)`

⇒ Let m₁ be the slope of the line segment joining (1, 5) and (3, −5):

`m_1 = (-5 - 5)/(3 - 1)`

`m_1 = (-10)/2`

∴ m₁ = −5

⇒ Let m₂ be the slope of the line segment joining C(2, −1) and D(−1, 9):

`m_2 = (2 - 3)/(-5 - 0)`

`m_2 = (-1)/-5`

∴ `m_2 = 1/5`

Here, two lines are perpendicular if m₁ × m₂ = −1:

`(-5) xx (1/5) = -1`

Since the product of the slopes is −1 the line segment joining (1, 5) and (3, −5) is perpendicular to the line segment joining the points (0,3) and (−5, 2).