If you were given two points, then you can safely say that they lie on one straight line as through any two points it is possible to draw a straight line. But how to find out whether all points on a straight line lie if points three, four or more? It is possible to prove accessory of points of one straight line in several ways.
It is required to you
- The points set by coordinates.
Instruction
1. If you were given points with coordinates (h1, u1, z1), (h2, u2, z2), (h3, u3, z3), find the straight line equation, using coordinates of any two points, for example, the first and second. For this purpose substitute the corresponding values in the straight line equation: (x-h1) / (h2-h1)= (at-u1) / (u2-u1)= (z-z1)/(z2-z1). If one of denominators is equal to zero, just equate numerator to zero.
2. To find the straight line equation, knowing two points with coordinates (h1, u1), (h2, u2), even more simply. For this purpose substitute values in a formula (x-h1) / (h2-h1)= (at-u1) / (u2-u1).
3. Having received the equation of the straight line passing through two points substitute values of coordinates of the third point in it instead of variables x and at. If equality turned out true, all three points mean lie on one straight line. In the same way you can check accessory of this straight line of other points.
4. Check accessory of all points of a straight line, having checked equality of tangents of tilt angles of the pieces connecting them. For this purpose check whether there will be true an equality (h2-h1) / (h3-h1)= (u2-u1) / (u3-u1)= (z2-z1)/(z3-z1). If one of denominators is equal to zero, then for accessory of all points of one straight line the condition h2-h1= h3-h1, u2-u1= u3-u1, has to be satisfied by z2-z1=z3-z1.
5. One more way to check accessory of three points of a straight line – consider the area of a triangle which they form. If all points lie on a straight line, then its area will be equal to zero. Substitute values of coordinates in a formula: S=1/2 ((h1-h3) (u2-u3) (-h2-h3) (u1-u3)). If after all calculations you received zero - means, three points lie on one straight line.
6. To find a solution of a task in the graphic way, construct the coordinate planes and find points on the specified coordinates. Then draw a straight line through two of them and continue to the third point, look whether it will pass through it. Consider, this way is suitable only for the points set on the plane with coordinates (x, s) if the point is set in space and has coordinates (x, at, z), such way is inapplicable.