Question

Line AB and line BC form a right angle at point B. If A = (2, 5) and B = (4, 4), what is the equation of line BC?

Answer 1

y = 2x - 4

Explanation:

To solve this problem, we must first calculate the slope of the line AB using the formula:

`\boxed{m = (y_2 - y_1)/(x_2 - x_1)}`

where:

m ⇒ slope of the line

(x₁, y₁), (x₂, y₂) ⇒ coordinates of two points on the line

Therefore, for line AB with points A = (2, 5) and B = (4, 4) :

`m_(AB) = (5 - 4)/(2 - 4)`

⇒
`m_(AB) = (1)/(-2)`

⇒
`m_(AB) = -(1)/(2)`

Next, we have to calculate the slope of the line BC.

We know that the product of the slopes of two perpendicular lines is -1.

Therefore:

`m_(BC) * m_(AB) = -1` [Since BC and AB are at right angles to each other]

⇒
`m_(BC) * -(1)/(2) = -1`

⇒
`m_(BC) = -1 / -(1)/(2)` [Dividing both sides of the equation by -1/2]

⇒
`m_(BC) = \bf 2`

Next, we have to use the following formula to find the equation of line BC:

`\boxed{y - y_1 = m(x - x_1)}`

where (x₁, y₁) are the coordinates of a point on the line.

Point B = (4, 4) is on line BC, and its slope is 2. Therefore:

`y - 4 =2 (x - 4)`

⇒
`y - 4 = 2x - 8` [Distributing 2 into the brackets]

⇒
`y = 2x-4`

Therefore, the equation of line BC is y = 2x - 4.