Describe a point

distance

Distance sounds quite normal at first: the distance between two locations.

A Maps app shows you the shortest route between 2 cities that you would drive with a car. That's the blue line on the map.



Mathematically, however, “distance” always means the shortest way. Colloquially, this would be the "beeline" between 2 cities. That's the black line on the map.

Mathematically, “distance” means the shortest connection between two locations.


In math, point-to-point and point-to-line distances are interesting.

In your everyday language you may sometimes use: "Take the shortest distance ..." This is mathematically doubled. The distance is already the shortest connection.

Distance point to point

You determine the distance between 2 points by connecting the two points with a line.



You cannot use a zigzag line for the distance.

The distance between 2 points $$ A $$ and $$ B $$ is the length of the segment $$ bar (AB) $$. For the length of $$ bar (AB) $$ you also write $$ | AB | $$.

Distance point to line

But what is the distance between a point P and a straight line?



You can draw different connecting lines from point P to the straight line.



You are also looking for the shortest connection here. This is the black route.

The black line thus marks the distance from P to the straight line. she stands perpendicular to the straight line.



It is sufficient if you draw the vertical line between the point and the straight line to determine the distance.


You draw the distance between point and straight line using a vertical line through point P.

kapiert.decan do more:

  • interactive exercises
    and tests
  • individual classwork trainer
  • Learning manager

Small repetition: Draw vertical lines

How to draw the perpendicular to a straight line through a point:



Place the set square with the center line on the straight line. Slide the set square until you reach the point. Draw the vertical.




For the distance it is enough to draw the route.

If you want to check whether you really have drawn a vertical line, you can place the tip of the set square at a right angle. If the tip fits exactly, there is also a 90 ° angle. In a pinch, you can do this with a sheet of paper.

Measure distance

How big is the distance now? You already know that: you measure distances with a ruler or set square.

With ruler

You place the ruler with the 0 at the starting point of the route. It is important that you do not place the example ruler below with the edge at the starting point. The standard of this ruler does not begin there.



You place the 0 at the point from which you measure.



Here the distance from the point to the straight line is 4.5 cm.

With set square

You also place the set square with the 0 at the starting point of the route. Only here is the 0 in the middle of the longest side of the set square. Then you can read off the length of the route.



It doesn't matter whether you measure from point to line or from line to point. The result is the same, otherwise you have measured yourself.


Here the distance from P to the straight line is 4.5 cm.



With the set square you can even measure and draw at the same time. To do this, place the set square with the center line on the straight line.


Some rulers do not have 0. The first number that is there is 1. This is because the measuring scale of the ruler begins at the very edge.

What do you need the distance for?

Here you can see a few examples of the distance in everyday life.

aviation

The aircraft is constantly checking how far the aircraft is from the ground. Measuring instruments measure the distance. So the pilot can correct the wrong distance immediately. This is important so that there are no collisions in the air and so that the aircraft always flies high enough.

shipping

In the ship, measuring instruments check the distance to the sea floor. This is to prevent the ship from falling into a shoal. Shallows are areas where the sea floor is higher than usual. This is more common on the coast.

The shortest distance to land is also determined from the ship. In this way, the captain can keep the route as short as possible.

Cross streets

Did your parents often say you shouldn't cross the street at an angle? You should take the shortest, i.e. the fastest, route across the street. That’s the safest. The shortest way is the mathematical distance to the roadside. You should go across the street at a right angle.

kapiert.decan do more:

  • interactive exercises
    and tests
  • individual classwork trainer
  • Learning manager

Examples from mathematics

Mirror image

If you want to draw a mirror image, you can do that with the distance. Place the set square with the center line on the mirror axis. Measure the distance between the points to be mirrored on one side and enter the points on the other side of the center line at the same distance.


Height of figures

If you want to measure the height in a figure, this is the distance from a point to a line in the figure.



Example:
You determine the height of the triangle on side c by placing the set square with the center line on side c. Now push the set square until you reach point C. Then you can measure the distance.