The discussion focuses on calculating the perpendicular distance from the point P(1, 3) to the line defined by the equation y = (x/2) - 5. Participants emphasize using the distance formula and the properties of perpendicular lines, specifically that their gradients multiply to -1. The gradient of the line perpendicular to y = (x/2) - 5 is determined to be -2, leading to the equation y = -2x + 5. To find the intersection point of the two lines, users are instructed to set the equations equal and solve for x.
PREREQUISITESStudents, educators, and anyone interested in mastering coordinate geometry, particularly those focused on understanding distances and relationships between lines in a Cartesian plane.
RTCNTC said:Use the distance formula to find the perpendicular distance from P(1, 3) to the line y = (x/2) - 5.
Any ideas on how to get started?
Prove It said:Perpendicular lines have gradients that multiply to give -1, so the gradient of the line you are looking for is -2. You know that (1, 3) lies on this line, so
$\displaystyle \begin{align*} y - 3 &= -2 \left( x - 1 \right) \\ y - 3 &= -2\,x + 2\\ y &= -2\,x + 5 \end{align*}$
And now you want to know where $\displaystyle \begin{align*} y = -2\,x + 5 \end{align*}$ and $\displaystyle \begin{align*} y = \frac{x}{2} - 5 \end{align*}$ intersect, so that you can work out the distance between this point and P(1, 3).
RTCNTC said:To find where the two lines meet, do I set the equations equal to each other?
MarkFL said:Yes, set:
$$-2x+5=\frac{x}{2}-5$$
and solve for $x$ to get the $x$-coordinate of the intersection point. Then plug this value for $x$ into either line (doesn't matter which as they will give the same $y$ value) to get the $y$-coordinate. :D