Download Aim #5: What are the properties of points, lines, and planes in three

Aim #5:
CC Geometry H
What are the properties of points, lines, and planes in three dimensional space?
Do Now: The undefined terms in geometry are: point, line, and plane.
Point: indicates place or position and is usually named by a capital letter.
Plane: a set of points that form a flat surface extending indefinitely in all
directions.
Line: a continuous set of points that extend indefinitely in two opposite directions.
Given MN that lies in plane p:
a) Draw RS that is perpendicular to p at point N.
b) Draw XY that is perpendicular to p at M.
c) How are RS and XY related?
d) Draw a plane q that is parallel to p.
e) True or false?: (1) XY and RS are perpendicular to q.
(2) Points M, N, X, and Y are coplanar.
M
p
N
1) The diagram shown is a three-dimensional right rectangular prism.
a. Name all lines parallel to AA'.
D'
A'
Name all lines parallel to AB.
Name all lines parallel to BC.
C'
B'
b. Name all pairs of parallel faces:
D
A
c. Name all lines perpendicular to B'B.
B
d. Name two lines that neither intersect nor are parallel (skew lines).
e. Do AB and A'B' lie in the same plane? If so, name the plane.
f. AB ∩ B'C' =
g. Will AC ever intersect the top face? Why or why not?
h. Is there another line that we could draw through B that would also be
perpendicular to the base?
C
Properties of Points, Lines, and Planes in Three-Dimensional Space
Property
Diagram
1. Two points determine a line/segment/ray.
2. Three points determine a plane.
3. Two lines are either intersecting, parallel,
or skew.
4. Through a given point not on a line l,
there is exactly one line parallel to l.
5. Two planes either intersect in a ______,
or they are parallel.
6. A line l is perpendicular to a plane if, at their
point of intersection, the plane contains two
lines perpendicular to l. (Every line in the plane
intersecting l is perpendicular to l.)
7. Two planes perpendicular to the same
line are ________________.
8. Two lines perpendicular to the same
plane are ________________.
9. Two segments connecting parallel planes
are ____ and ____ if they are perpendicular
to the planes.
10. The distance from a point to a plane
is the length of a ______________
segment from the point to the plane.
l
2) Determine the length of AC' given the dimensions 3 x 4 x 5.
D'
A'
C'
B'
D
A
5
3
B
4
C
3) Indicate whether each statement is always true (A), sometimes true (S), or
never true (N).
a. If two lines are perpendicular to the same plane, the lines are parallel.
b. Two planes can intersect in a point.
c. Two lines parallel to the same plane are perpendicular to each other.
d. If a line meets a plane in one point, then it must pass through the plane.
e. A line and a point not on the line form more than one plane.
4)
In the prism to the left, MA MS, MA MH. Lines
that appear to be parallel are parallel. Which plane is
perpendicular to MA?
(1) MAT
(2) HES
(3) MSP
5) A base of the 3-dimensional figure to the right is a regular
pentagon. If a plane slices through this figure parallel to the
base, the cross-section formed by the slice is a ____________.
6) In the diagram, line k is perpendicular to plane P at point T.
Which statement is true?
(a) Any point in plane P also will be on line k.
(b) Only one line in plane P will intersect line k.
(c) All planes that intersect plane P will pass through T.
(d) Any plane containing line k is perpendicular to plane P.
(4) SPL
7) Use the diagram to the right for a-e:
a) What can be concluded about the relationship
between line l and plane P? Why?
b) What can be concluded about the relationship
between planes P and Q? Why?
c) What can be concluded about the relationship
between lines l and m? Why?
d) What can be concluded about segments AB and CD?
e) Line j lies in Plane P, and line i lies in plane Q. What can be concluded about the
relationship between lines i and j?
8) Which group of points is not coplanar based on the picture below?
(1) D,A,F,E
(2) F,G,B,A
(3) E,F,G,H
(4) G,B,F,D
9) Lines m and n intersect at point A. Line k is perpendicular to both lines m and
n at point A. Which statement must be true?
(1) Lines m, n, and k are in the same plane.
(2) Lines m and n are in two different planes.
(3) Lines m and n are perpendicular to each other.
(4) Line k is perpendicular to the plane containing lines m and n.
10. Consider the right hexagonal prism whose bases are regular hexagonal regions.
The top and the bottom hexagonal regions are called the base faces, and the side
rectangular regions are called the lateral faces.
a. List a plane that is || to plane C'D'E'.
b. List all planes shown that are not || to plane
CDD'.
c. Name a line perpendicular to plane ABC.
d. Explain why AA' = CC'.
e. Is AB || to DE? Explain.
f. Is AB || to C'D'? Explain.
g. Is AB || to D'E'? Explain.
h. If line segments BC' and C'F' are perpendicular then is BC' perpendicular to
plane C'A'F'? Explain.
i. One of the following statements is false. Identify which statement is false and
explain why.
a) BB' is perpendicular to B'C'.
b) EE' is perpendicular to EF.
c) CC' is perpendicular to E'F'.
d) BC is || to F'E'.
Name ______________________
HW #5
CC Geometry H
Date ______________________
A'
1) Use the rectangular prism to answer a-f.
D'
a. To which edges is BB' perpendicular?
b. Name a line that intersects plane ABB'A'.
Name a line that will not intersect.
C'
B'
A
c. Are AB and D'C' coplanar? If yes, shade the plane.
D
300
4
d. Can lines AB and B'C' lie in the same plane?
B
e. plane ABCD ∩ plane C'D'DC = ________
f. Given CC' = 6 and AB = 4, find BC in simplest radical form.
2) Indicate whether each statement is always true (A), sometimes true (S),
or never true (N).
a. Skew lines can lie in the same plane.
b. If two lines are parallel to the same plane, the lines are parallel.
c. If two planes are parallel to the same line, they are parallel to each other.
d. If two lines do not intersect, they are parallel.
3) In the right triangular prism shown at the right, which planes are parallel?
(1) ECBF and DABF
(2) EDAC and ECBF
(3) ACB and DABF
(4) DEF and ACB
6
4) Choose all that are true.
a. If two planes are parallel to a third plane, the two planes are parallel.
b. If two planes are perpendicular to a third plane, then the two planes are ll.
c. If a plane intersects two ll planes, then the intersections are two ll lines.
d. If a line is perpendicular to a plane, then every plane containing the line is
perpendicular to the given plane.
5) If two distinct lines are perpendicular to the same plane, then the lines are
(1) collinear
(2) congruent
(3) coplanar
(4) consecutive
6) If three planes intersect as shown, the intersection forms
(1) a line
(2) two lines
(3) a fourth plane
(4) a rectangle
C
7) The diagram shows a right rectangular prism determined by vertices A, B, C, D,
E, F, G, and H. Square ABCD has sides of length 5 and AE = 9. Find DF to the
nearest tenth. (Hint: find AF first)
8) In the following figure, ΔABC is in plane P, ΔDEF is in plane Q, and BCFE is a
rectangle. Which of the following statements are true?
a. BE is perpendicular to plane Q.
b. BF = CE.
c. Plane P is parallel to plane Q.
d. ΔABC ≅ ΔDEF.
d. AE = AF.
Review: Find the shaded area to the nearest tenth:
3
6
3
6