Find the equation of line AB, given that it is perpendicular to line CD and passes through point A(0, −1)…

Question

Find the equation of line AB, given that it is perpendicular to line CD and passes through point A(0, −1). Points C and D are (0, 4) and (6, 0) respectively.

ScanSolve AI Tutor · Accepted Answer

To find the equation of line AB, we need its slope and a point it passes through.

Identify the coordinates of the given points.
From the diagram, point C is $(0, 4)$ and point D is $(6, 0)$ .
Point A is on the y-axis at $y = -1$ , so its coordinates are $(0, -1)$ .
Calculate the slope of line CD.
The slope $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $m = \frac{y_2 - y_1}{x_2 - x_1}$ .
For line CD, using C $(0, 4)$ and D $(6, 0)$ :
$m_{CD} = \frac{0 - 4}{6 - 0} = \frac{-4}{6} = -\frac{2}{3}$
Determine the slope of line AB.
The problem states that AB is perpendicular to CD. For two perpendicular lines, the product of their slopes is $-1$ .
$m_{AB} \cdot m_{CD} = -1$ $m_{AB} \cdot \left(-\frac{2}{3}\right) = -1$ $m_{AB} = \frac{-1}{-\frac{2}{3}} = \frac{3}{2}$
Find the equation of line AB.
We have the slope $m_{AB} = \frac{3}{2}$ and a point A $(0, -1)$ that line AB passes through. We can use the point-slope form of a linear equation, $y - y_1 = m(x - x_1)$ .
$y - (-1) = \frac{3}{2}(x - 0)$ $y + 1 = \frac{3}{2}x$ $y = \frac{3}{2}x - 1$
The equation of line AB is $\boxed{\mathbf{y = \frac{3}{2}x - 1}}$ .

Find the equation of line AB, given that it is perpendicular to line CD and passes through point A(0, -1). Points C and D are (0, 4) and (6, 0) respectively.

Step-by-step solution

Identify the coordinates of the given points.

Calculate the slope of line CD.

Determine the slope of line AB.

Find the equation of line AB.

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