Question

name a point that is collinear with points e and h. name a point that is coplanar with points d, a and b. name the intersection of planes aeh and fbe. complete the sentence with always, sometimes, never: a) a line ________ has endpoints. b) a line and a point ______ intersect. c) two planes ______ intersect in a lines. d) any three points not on the same lines ________ determine a plane. provide real - world examples of: 1) point: 2) line 3) plane 4) intersection of two planes explain: why a four - legged chair may rock from side to side even if the floor is level. would a three - legged chair on the same level floor rock from side to side? describe and correct the error. ad and ac are opposite rays. modeling real life: when two trucks traveling in different directions approach an intersection at the same time, one of the trucks must change its speed or direction to avoid a collision. two airplanes, however, can travel in different directions and cross paths without colliding. explain how this is possible.

Explanation:

Step1: Recall collinear points concept

Points on the same line as E and H are collinear. Point J is on the line segment EH, so it is collinear with E and H.

Step2: Recall coplanar points concept

Points on the same plane as D, A and B are coplanar. Point C is on the same face (plane) as D, A and B of the cube.

Step3: Determine plane - intersection

The intersection of planes AEH and FBE is line BE since it is the common line between the two planes.

Step4: Complete sentences with geometric facts

a) A line never has endpoints as it extends infinitely in both directions.
b) A line and a point sometimes intersect. If the point is on the line, they intersect; if not, they don't.
c) Two planes always intersect in a line (assuming they are not parallel).
d) Any three points not on the same line always determine a plane.

Step5: Provide real - world examples

1. Point: The tip of a pencil.
2. Line: The edge of a ruler.
3. Plane: The surface of a table.
4. Intersection of two planes: The intersection of two adjacent walls in a room (the corner line).

Step6: Explain chair - stability

A four - legged chair may rock from side to side on a level floor because the four points (legs) may not all lie in the same plane. A three - legged chair on a level floor will not rock because any three non - collinear points determine a unique plane.

Step7: Correct ray error

The error is that for two rays to be opposite rays, they must have the same endpoint and extend in opposite directions. Here, $\overrightarrow{AD}$ and $\overrightarrow{AC}$ do not extend in opposite directions. They share the same endpoint A but are not in a straight - line opposite orientation.

Step8: Explain truck and airplane situation

Trucks travel on a two - dimensional surface (the road), so when they approach an intersection, they are in the same plane and may collide. Airplanes travel in three - dimensional space. They can have different altitudes, so even if their paths cross in the horizontal plane, they can avoid collision by being at different heights.

Answer:

1. Point J is collinear with E and H.
2. Point C is coplanar with D, A and B.
3. The intersection of planes AEH and FBE is line BE.
4. a) never

b) sometimes
c) always
d) always

1. 1) The tip of a pencil.
2. The edge of a ruler.
3. The surface of a table.
4. The intersection of two adjacent walls in a room (the corner line).
5. A four - legged chair may rock because the four points (legs) may not all lie in the same plane. A three - legged chair will not rock as any three non - collinear points determine a unique plane.
6. The error is that $\overrightarrow{AD}$ and $\overrightarrow{AC}$ do not extend in opposite directions. For rays to be opposite, they must have the same endpoint and extend in opposite directions.
7. Trucks travel on a two - dimensional surface (road), so they may collide at an intersection. Airplanes travel in three - dimensional space and can have different altitudes to avoid collision even if their horizontal paths cross.