Question

Without using Pythagoras theorem, show that the points A(0, 4), B(1, 2) and C(−4, 2) are the vertices of a right-angled triangle.

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 A(0, 4) and B(1, 2):

`m_1 = (2 - 4)/(1 - 2)`

`m_1 = (-2)/1`

∴ m₁ = −2

⇒ Let m₂ be the slope of the line segment joining B(1, 2) and C(−4, 2):

`m_2 = (2 - 2)/(-4 - 1)`

`m_2 = 0/-5`

∴ m₂ = 0

⇒ Let m₃ be the slope of the line segment joining A(0, 4) and C(−4, 2):

`m_3 = (2 - 4)/(-4 - 0)`

`m_3 = (-2)/-4`

∴ `m_3 = 1/2`

Now, checking if the product of any two slopes equals −1:

`m_1 xx m_3 = (-2) xx (1/2)`

= −1

Since the product of the slopes of AB and AC is −1, the sides AB and AC are perpendicular to each other. Therefore, ΔABC  is a right-angled triangle with the right angle at vertex A.