How can I show that the lines between sets of points are perpendicular?

For example, A(6/5, 19/5), B(2, 9/5), C(5, 3). Show that the line segments AB and BC are perpendicular.

Before we begin the question, it is a good idea to remind ourselves of the definition of two lines being perpendicular.

Two lines are perpendicular (at right angles to eachother), if the gradients of the two lines multiply to give -1.

It is also worth reminding ourselves how to calculate the gradient of a line, given two points.

If the two points are (x1, y1), (x2, y2), m (the gradient) = (y2 - y1)/(x2-x1)

Back to the question...

We will need to show that the gradient of the line AB multiplied by the gradient of the line BC gives -1.

Finding the gradient of AB

A(6/5, 19/5), B(2, 9/5)

m_AB = ((9/5) - 19/5)/(2 - (6/5)) = -2/(4/5) = -5/2

Finding the gradient of BC

B(2, 9/5), C(5,3)

m_BC = (3 - (9/5))/(5 - 2) = (6/5)/3 = 2/5

Now we have found the gradients, we will multiply them together and see if we get -1. If this is case, we have shown that the lines are perpendicular.

m_AB * m_BC = (-5/2) * (2/5) = -1

As the gradients multiply to get -1, we have shown that the line segments AB and BC are perpendicular.

Here are a few more questions to test your understanding:

1. A(1, 7), B(2, 3), C(6, 4). Show that the line segments AB and BC are perpendicular.

2. Show that the lines y = (1/3)x + 2 and y = -3x+7 are perpendicular (Hint: Consider the general form of the equation of the line)

3. Show that the tangent to y = x² + 2x at x = 2 is perpendicular to the line segment AB, where A is the point (2, 3) and B is the point (8, 2). (Hint: Use differentiation to find the gradient of the tangent)