the golden rule
This single principle, if remembered, allows you to solve all distance problems.
Perpendicular
What does the point represent in the classic example of distance to a plane?
Drone
If a line and a plane are parallel, where on the line can you take a point to measure the distance?
Any point
Can skew lines lie in the same plane?
No
"The floor and ceiling in a room" is an example of two such objects.
Parallel planes
What kind of line (at what angle) is always used to measure the shortest distance?
90°
What is the name of the segment whose length equals the desired distance from a point to a plane?
AH (perpendicular segment)
Why is the distance between parallel planes or a line and a plane the same everywhere?
Parallelism → constant distance
What two properties define skew lines?
Not parallel and do not intersect
How is the distance between them indicated on the schematic drawing of the ceiling and floor?
Arrow labeled “distance”
Why can't a slanted or diagonal line be used to measure distance?
Not the shortest path
What is the "memory trick" for finding this distance?
Straight drop, not slide
To find the distance between two parallel planes, you need to draw a perpendicular from... where?
From any point on one plane to the other
What is the name of the unique line that can be drawn between two skew lines to measure the distance between them?
Common perpendicular
If one line is on a wall and another is on the floor, and they don't intersect, what are their possible mutual positions?
Skew lines
State the "Golden Rule" for finding distances in space.
Perpendicular = shortest distance
If several lines are drawn from a point to a plane, the shortest one will form this angle with the plane.
Right angle
In the bridge over the river example: what is the line and what is the plane?
Bridge = line, river = plane
In the classic example, one line lies on the floor. Where does the second one lie?
On a wall
To find the distance between skew lines, you need to find the length of the line that is... to them (complete the definition).
Perpendicular to both
This rule applies to all four main types of problems. Name them.
Point–plane, line–plane (parallel), two parallel planes, skew lines
What two things are needed to calculate the shortest distance from a point to a plane in 3D geometry?
You need 1) the coordinates of the point and 2) the equation of the plane.
How is the problem of finding the distance between two parallel planes related to the problem of finding the distance from a point to a plane?
Equivalent to “point–plane” problem
This perpendicular is common to both skew lines. What does that mean?
Perpendicular to both lines
Describe the complete plan of action for finding the distance between two parallel planes in no more than 3 steps.
1. Take a point on plane 1 → 2) Drop perpendicular to plane 2 → 3) Measure length