How to Find the Perpendicular Distance from a Point to a Line?

In summary, the perpendicular distance from P(1, 3) to the line y = (x/2) - 5 is -2. The intersection point of these two lines is (x=-2, y=5), so the distance between the point and P(1, 3) is 5.
  • #1
mathdad
1,283
1
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?
 
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  • #2
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?
Try starting with this.

-Dan
 
  • #3
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?

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).
 
  • #4
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).

To find where the two lines meet, do I set the equations equal to each other?
 
  • #5
RTCNTC said:
To find where the two lines meet, do I set the equations equal to each other?

Yes, set:

\(\displaystyle -2x+5=\frac{x}{2}-5\)

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
 
  • #6
MarkFL said:
Yes, set:

\(\displaystyle -2x+5=\frac{x}{2}-5\)

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

I can do it. Thanks.
 

FAQ: How to Find the Perpendicular Distance from a Point to a Line?

1. What is perpendicular distance?

Perpendicular distance refers to the shortest distance between a point and a line or a plane. It is the distance that is perpendicular (at a right angle) to the line or plane.

2. How is perpendicular distance calculated?

The perpendicular distance can be calculated using the formula d = |ax + by + c| / √(a² + b²), where (x,y) is the coordinates of the point, and a, b, and c are the coefficients of the line or plane's equation.

3. What is the significance of perpendicular distance in geometry?

In geometry, perpendicular distance is important because it helps to determine the shortest distance between a point and a line or plane. It is also used in various theorems and geometric constructions.

4. Can perpendicular distance be negative?

Yes, perpendicular distance can be negative. This occurs when the point lies on the opposite side of the line or plane from the origin of the coordinate system.

5. How is perpendicular distance used in real life?

Perpendicular distance has various applications in real life, such as in architecture and engineering for determining the shortest distance between two points or for creating precise angles. It is also used in navigation and surveying to calculate distances and angles between points.

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