Three-Dimensional Space

New York State Common Core Math Geometry, Module 3, Lesson 5

Worksheets for Geometry

Student Outcomes

• Students describe properties of points, lines, and planes in three-dimensional space.

Three-Dimensional Space

Classwork

Exercise

The following three-dimensional right rectangular prism has dimensions 3 × 4 × 5. Determine the length of 𝐴𝐶′. Show a full solution.

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Exploratory Challenge

1. Two points 𝑃 and 𝑄 determine a distance 𝑃𝑄, a line segment 𝑃𝑄, a ray 𝑃𝑄, a vector 𝑃𝑄, and a line 𝑃𝑄.
2. Three non-collinear points 𝐴, 𝐵, and 𝐶 determine a plane 𝐴𝐵𝐶 and, in that plane, determine a triangle 𝐴𝐵𝐶.
3. Two lines either meet in a single point, or they do not meet. Lines that do not meet and lie in a plane are called parallel. Skew lines are lines that do not meet and are not parallel.
4. Given a line ℓ and a point not on ℓ, there is a unique line through the point that is parallel to ℓ.
5. Given a line ℓ and a plane 𝑃, then ℓ lies in 𝑃, ℓ meets 𝑃 in a single point, or ℓ does not meet 𝑃, in which case we say ℓ is parallel to 𝑃. (Note: This implies that if two points lie in a plane, then the line determined by the two points is also in the plane.)
6. Two planes either meet in a line, or they do not meet, in which case we say the planes are parallel.
7. Two rays with the same vertex form an angle. The angle lies in a plane and can be measured by degree.
8. Two lines are perpendicular if they meet, and any of the angles formed between the lines is a right angle. Two segments or rays are perpendicular if the lines containing them are perpendicular lines.
9. A line ℓ is perpendicular to a plane 𝑃 if they meet in a single point, and the plane contains two lines that are perpendicular to ℓ, in which case every line in 𝑃 that meets ℓ is perpendicular to ℓ. A segment or ray is perpendicular to a plane if the line determined by the ray or segment is perpendicular to the plane.
10. Two planes perpendicular to the same line are parallel.
11. Two lines perpendicular to the same plane are parallel.
12. Any two line segments connecting parallel planes have the same length if they are each perpendicular to one (and hence both) of the planes.
13. The distance between a point and a plane is the length of the perpendicular segment from the point to the plane. The distance is defined to be zero if the point is on the plane. The distance between two planes is the distance from a point in one plane to the other.

Lesson Summary

SEGMENT: The segment between points 𝐴 and 𝐵 is the set consisting of 𝐴, 𝐵, and all points on ⃡𝐴𝐵 between 𝐴 and 𝐵. The segment is denoted by 𝐴𝐵, and the points 𝐴 and 𝐵 are called the endpoints.

LINE PERPENDICULAR TO A PLANE: A line 𝐿 intersecting a plane 𝐸 at a point 𝑃 is said to be perpendicular to the plane 𝐸 if 𝐿 is perpendicular to every line that (1) lies in 𝐸 and (2) passes through the point 𝑃. A segment is said to be perpendicular to a plane if the line that contains the segment is perpendicular to the plane.

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