The discussion revolves around finding the perpendicular distance from a point P(1, 3) to the line defined by the equation y = (x/2) - 5. Participants explore methods to calculate this distance, focusing on the use of the distance formula and the intersection of lines.
Participants generally agree on the approach to finding the perpendicular distance, but there is no consensus on the specific steps or the final calculations involved.
The discussion does not resolve the mathematical steps required to find the intersection point or the final distance, leaving some assumptions and calculations unaddressed.
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