In mathematics, both algebra and geometry set the task of finding the distance to a point or line from a given object. It is found in completely different ways, the choice of which depends on the initial data. Consider how to find the distance between given objects in different conditions.
Using measuring tools
At the initial stage of mastering mathematical science, they teach how to use elementary tools (such as a ruler, protractor, compass, triangle, and others). Finding the distance between points or lines with their help is not difficult at all. It is enough to attach the scale of divisions and write down the answer. One has only to know that the distance will be equal to the length of the straight line that can be drawn between the points, and in the case of parallel lines, the perpendicular between them.
Using theorems and axioms of geometry
In high school, they learn to measure distance without the help of special devices or graph paper. This requires numerous theorems, axioms and their proofs. Often the problems of how to find the distance come down toforming a right triangle and finding its sides. To solve such problems, it is enough to know the Pythagorean theorem, the properties of triangles and how to transform them.
Points on the coordinate plane
If there are two points and given their position on the coordinate axis, how to find the distance from one to the other? The solution will include several steps:
- Connect the points with a straight line, the length of which will be the distance between them.
- Find the difference between the coordinates of the points (k;p) of each axis: |k1 - k2|=q 1 and |p1 - p2|=d2 (values are taken modulo, because the distance cannot be negative).
- After that, we square the resulting numbers and find their sum: d12 + d2 2
- The final step is to extract the square root of the resulting number. This will be the distance between the points: d=V (d12 + d2 2).
As a result, the whole solution is carried out according to one formula, where the distance is equal to the square root of the sum of squares of the difference in coordinates:
d=V(|k1 - k2|2+|r 1 - p2|2)
If the question arises of how to find the distance from one point to another in three-dimensional space, then the search for an answer to it will not be much different from the above. The decision will be made according to the following formula:
d=V(|k1 -k2|2+|p1 - p2 |2+|e1 - e2|2)
Parallel lines
The perpendicular drawn from any point lying on one straight line to the parallel will be the distance. When solving problems in a plane, it is necessary to find the coordinates of any point of one of the lines. And then calculate the distance from it to the second straight line. To do this, we bring them to the general equation of a straight line of the form Ax + Vy + C \u003d 0. It is known from the properties of parallel lines that their coefficients A and B will be equal. In this case, you can find the distance between parallel lines using the formula:
d=|C1 - C2|/V(A2 + B 2)
Thus, when answering the question of how to find the distance from a given object, it is necessary to be guided by the condition of the problem and the tools provided for its solution. They can be both measuring devices, and theorems and formulas.
