The distance from a point to the plane equals to length of a perpendicular which is lowered on the plane from this point. All further geometrical constructions and measurements are based on this definition.
It is required to you
- - ruler;
- - drawing triangle with a right angle;
- - compasses.
Instruction
1. To find distance from a point to the plane: • draw the straight line perpendicular to this plane through this point; • find the perpendicular basis - a straight line point of intersection with the plane; • measure distance between the set point and the basis of a perpendicular.
2. For finding of distance from a point to the plane by methods of descriptive geometry: • choose any point on the plane; • draw through it two the straight lines (lying in this plane); • restore the perpendicular to the plane passing through this point (construct the straight line perpendicular to at the same time both crossed straight lines); • draw through the set point a straight line parallel, to the constructed perpendicular; • find distance between a point of intersection of this straight line with the plane and the set point.
3. If the provision of a point is set by its three-dimensional coordinates, and the provision of the plane – the linear equation, then to find distance from the plane to a point, use methods of analytical geometry: • designate point coordinates through x, y, z, respectively (x – an abscissa, y – ordinate, z – z-coordinate); • designate through And, In, With, the D parameters of the equation of the plane (And – parameter at an abscissa, In – at ordinate, With – at z-coordinate, D – the free member); • calculate distance from a point to the plane on a formula: s = | (Ax+By+Cz+D) / √ (A²+B²+C²) | where s is an oasstoyaniye between a point and the plane, | | - designation of absolute value (or the module) numbers.
4. Example. Find distance between a point And with coordinates (2, 3,-1) and the plane set by the equation: 7kh-6u-6z+20=0. Decision. Follows from statements of the problem that: x =2, at =3, z=-1, A=7, B=-6,C=-6,D=20. Substitute these values in the above-stated formula. It will turn out: s = | (7*2+(-6) *3+ (-6) * (-1)+20) / √ (7²+ (-6)²+ (-6)²) | = | (14-18+6+20)/11 | = 2. Answer: The distance from a point to the plane is equal to 2 (conventional units).