Three dimensional object with vocabulary, material explanation and sample questions

4. A.Position of Point, Line, and Plane in Polyhedral
Point, line and plane are main component to build polyhendral such as rectagural prism, cube,
pyramids and so on. We learn the meaning of point, line and plane and then discuss their position in
polyhendral.
1. The Definition of Point, Line and Plane
a. Point
A point is determined by its position and does not have value. A point is notated as a dot and an
uppercase alphabet such A, B, C and so on.
A B
Point A Point B
b. Line
A line is a set of unlimited series of points. A line is called one-dimensioanal object since it only has
one measurement. A line is usually drawn with ends and called asegment of line (or just segment) and
notated in a lowercase alphabet, for example, line g, h, i. A segment is commonly notated by its end point,
for examples, segment AB, PQ.

5. Line g A segment AB B
c. Plane
A plane is defined as a set of points that have length and area, therefore planes are called two-
dimensioanal objects. A plane is notated using symbol like ∝, 𝛽, 𝛾, or its vertexes.
C D
(a) A (b) B
plane∝ plane ABCD
2. Axioms of Line and Plane
Axiom is a statement that is accepted as true without further proof or argument.. the following are
sereval axioms about point, line and plane.
Axiom 1:
Through any two different points can only be drawn a straight line.
A B
g

6. Axiom 2:
If a line and a plane have two intersection point, thenthe line ison the plane.
A B
Axiom 3:
Through any three different point which are not in a line can only be drawn a plane.
C
A
B
∝
∝

7. Axiom 4:
Through a point outside of a defined line can only be drawn a line which is parallel to the defined line.
g
h
A
3. Position Of a Point toword a Line
There are two possible positions of a point toword line, which are the point is either on the line or
outside the line.
a. A Point on the Line
A point is stated on a line if the point is passed by the line.
A
g
b. A Point Outside a Line
A point is outside a line if the point is not passed through by the line.
∝

8. A
g
4. Position Of a Point Toward a Plane
a. Point on a Plane
A point is on a plane if the point is passed by the plane.
V
A
b. A Point Outside a Plane
A point is outside a plane if the point is not passed by the plane.
V
A

9. A B
C
D
H
E
G
Example:
Given a cube ABCD.EFGH. Find the vertexes of the cube which are:
a. On line EG H G
b. On the outside line EG E F
c. On the plane ABCD
d. On the outside of the plane ACH
D C
A B
Answers:
a. Points on the line EG are points E and G
b. Points on the outside of line EG are points A,B, C, D, F and H
c. Points on the plane ABCD are points A, B, C and D
d. Points on the outside of plane BDG are points A, C, E, F and H
5. Position Of a Line Toward Other Lines
a. Two Line Intersect Each Other
Two line intersect each other if these lines are on a plane and have a point of intersection.
V
C’
T
g
P
h

10. b. Two Parallel Lines
Two line are parallel if these lines are on a plane and do not have a point intersection.
V
g
h
c. Two Lines Cross Over
Two lines cross over each other if these lines are not on the same plane or not form a plane.
b
g
6. Position of a Line Toward a Plane
The position of a line toward a plane may be the line is on the plane, the line is parallel to the plane or
the line intersects (cuts) the plane.

11. a. A Line On a Plane
A line ison the plane if the line and the plane have at least two points of intersection.
A B
g
b. A Line Parallel to a Plane
A line is parallel to a plane if they do not have any point of intersection.
g
c. A line Intersects a Plane
A line intersects a plane if they have at least a point of intersection.
g
A

12. Example:
Given a cube ABCD. EFGH. Find the cube which:
a. Intersects BD
b. Parallel to BC
c. Cross over EG
d. On face ABCD
e. Parallel to face ABCD, and
f. Intersects face ABCD
Answers:
a. Edge intersecting BD are AC, BC, AB, AD, BF, and DH
b. Edge which are parallel to BC are AD, EH, and FG
c. Edge which cross over EG are DH, BF, AD, BC, AB, and DC
d. Edge which are on face ABCD are AB, AD, BC, and CD
e. Edge which are parallel to face ABCD are EF, EH, FG, and GH
f. Edge intersecting face ABCD are AE, BF, CG, and DH
7. Of a Plane to Other Planes
Positions between two planes may be parallel, one is on the other or intersecting.
a. Two Parallel Planes
Planes V and W are parallel if these planes do not have any point of intersecting.
H G
C
F
E
D
A B

13. V
W
V
W
V
V
V
W
W
W
W
W
W
W
W
W
W
W
W
W
V
W
b. A Plane is on The Other Planes
Plane V and W are on aech other of every point on V is also on W, or line versa
c. Two Intersecting Planes
Plane V and W intersect each other if they have exactly only one line of intersecting, which is
called an intersecting line.
(V,W)
V
W
VW

14. Example:
In cube ABCD. EFGH, draw a point of intersection by line AG towards plane BDHF.
Answers:
Procedures:
1. Draw dialogonals of the base and top faces, which are AC, BC, EG, and FH
2. Draw line PQ connecting the point of intersection of the diagonals on the base and the stop
3. Draw line AG which intercepts line PQ on point R, point R is the point of intersection by line AG
toward plene BDHF
H
E
G
P
F
R
A B
C
D
Q

15. B. Drawing Polyhendral (3-D Drawing)
Before you learn ways to draw polyhedral (commonly called 3-D Drawing), you need several important
terms. Key terms in drawing polyhendral are:
1. Drawing Field
A drawing field is a plane that is used as the pad of the drawing. A drawing field is commonly notated
by α, β, and γ. In particular, it must face upfront to the eyes of the observer. A drawing paper or a writing
board can be used as drawing field. In picture 7.24 the drawing field is represented by plane α.
W V
T U
P
S R
Q

16. 2. Frontal Plane
A frontal plane is a plane that is parallel to a drawing field. In picture 7.24, planes SRVW and PQUT
are frontal planes. In particular, a frontal plane has exact measurement as the actual figure.
3. Frontal Line
Lines or edges on the frontal plane are called frontal lines. Based on the direction, there are two
frontal lines, namely horizontal frontal line and vertical frontal line. Lines PQ, TU, SR and WV are
horizontal frontal lines, while lines PT, QU, RV and SW are vertical frontal ones.
4. Orthogonal Plane
An orthogonal plane ia a plane which is horizontally or vertically perpendicular to frontal plane,
either to the front or to the back. In picture 7.24, PQRS and TUVW are horizontal orthogonal planes, while
PSTW and QRVU are vertical orthogonal planes.
5. Orthogonal Line
An orthogonal line is a which is line perpendicular to frontal plane. In picture 7.24, lines PS, QR,
UV and TW are orthogonal lines.
6. Orthogonal Ratio
An orthogonal ratio or projection ratio is ratio between the length of an orthogonal line in the figure
and in the actual. In picture 7.24, the orthogonal ratio is:

17. Orthogonal ratio =
𝑇ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑃𝑆 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑖𝑔𝑢𝑟𝑒
𝑇ℎ𝑒 𝑎𝑐𝑡𝑢𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑃𝑆
7. Oblique Angel
An oblique angel is an angel formed by horizontal frontal line and orthogonal line. This angel is
used to make a good perspective of 3-D drawings. For an example, in picture 7.24 sudut SPQ is drawn as
an acute angel, although the real angel is a right angel.
See the following example to understand how to make 3-D drawings.
Example:
Given a cube PQRS.TUVW whose edge length is 4 cm. Draw the cube if PRVT is frontal face, PR is
horizontal frontal line, oblique angel is 30° and orthogonal ratios is 1:2
T
W V
U
S R
P Q
30°

18. Answer:
Let the frontal planePRVT.
Since the edge length is 4 cm, then the length of diagonal of each face is PR = 4√2 cm.
Steps to draw the cube are:
1.Draw a square as the frontal plane PRVTwith PR = 4√2 cm and RV = 4cm.
2.Determine the mid point of PR which is O.
3.By point O, make an angle 30° toward OR.
4.Find the length OS = OQ =
1
2
x
1
2
QS
=
1
4
x 4√2 = √2
5.Draw face PQRS
6.Draw the other edges by connecting the corresponding vertexes.

19. C. Distance in Polyhedral
Distance between two objects is lenght of shortest segment connecting these objects. This segment is
perpendicular to the objects. The length is shown by a positive number.
1. Distance between Two Points
Distance between two points is the length of segment connecting these points. Distance between
points P and Q is the length of segment PQ, which is d. P Q
2. Distance between a Point and a Line
The distance between a point and line is the length of segment drawn from the point perpendicular to
the line. The distance of point P and line g is segment PQ, which is perpendicular to line g, and has length
d.
d

20. 3. Distance between a Point and Plane
The distance between a point and plane is the length of segment drawn from the point perpendicular to
the plane. The distance between point P and plane V is the length of segment PQ, which is perpendicular to
plane V and has length d.
4. Distance between Point Two Lines
The distance between two parallel or cross over lines is the length of segment perpendicular to both
lines. The distance between line g and line h is the length of segment PQ, which is perpendicular to line g
and h, and has length d.

21. 5. Distance between Line and Plane
The distance between a line and the parallel plane is the length of segment which is perpendicular to the
line line and the plane. The distance between line g and the plane V is the length of segment PQ which is
perpendicular to line g and plane V and has length d.
6. Distance between Two Plane
The distance between two planes is length of segment which is perpendicular to both planes. The
distance between plane V and W is the length of PQ, which is
perpendicular to plane V and plane W and has length d.

22. Example :
Given a cube ABCD. EFGH with edge length 8 cm. Point P, Q and R are
in the mid points of edges AB, BC, and plane ADHE respectively.
Find the distance between:
a. Point P and R
b. Point Q and R
Answers:
a. See that ∆𝑃𝐴𝑅 has a right angle on A.
𝐴𝑃 =
1
2
𝐴𝐵 = 4 𝑐𝑚
𝐴𝑅 =
1
2
𝐴𝐻 =
1
2
√𝐴𝐷2 + 𝐷𝐻2
=
1
2
√82 + 82 = 4√2
𝑃𝑅 = √𝐴𝑃2 + 𝐴𝑅2
= √42 + (4√2)
2
= √48 = 4√3

23. So, the distance between point P and point R is 4√3 cm.
b. See that ∆𝑄𝑅𝑆 has a right angle on S.
𝑄𝑆 = 8 𝑐𝑚 𝑎𝑛𝑑 𝑅𝑆 =
1
2
𝐴𝐸 = 4 𝑐𝑚
𝑄𝑅 = √𝑄𝑆2 + 𝑅𝑆2 = √82 + 42 = √80 = 4√5
So, the distance between point Q and point R is 4√5 cm.
Example :
Given a cube ABCD. EFGH with edge length 6 cm. Point P and Q are the centre point of planes EFGH
and ABCD respectively. Find the distance between lines QF and DP.

24. Answers:
See the figure beside!
𝐵𝐷 = 6√2 𝑐𝑚
𝐷𝑄 =
1
2
𝐵𝐷 = 3√2 𝑐𝑚
Because ∆𝐷𝑃𝑄 has a right angle on Q, then
𝐷𝑝 = √𝐷𝑄2 + 𝑄𝑃2
= √(3√2)
2
+ 62
= √54 = 3√6
Area of ∆𝐷𝑃𝑄 =
1
2
. 𝐷𝑄. 𝑃𝑄 =
1
2
. 𝐷𝑃. 𝑄𝑅
Thus
1
2
. 𝐷𝑄. 𝑃𝑄 =
1
2
. 𝐷𝑃. 𝑄𝑅
𝑄𝑅 =
𝐷𝑄.𝑃𝑄
𝐷𝑃
=
3√2 . 6
3√6
= 2√3
So, the distance between lines QF and DP is segment QR = 2√3 𝑐𝑚

25. D. Angle Size on Polyhedral
1. Angle Formed by Two Lines in Space
The theorem of two equal angles in Plane Geometry (Two Dimensional Geometry) can be used to
determine angle size formed by two lines in space.
a. Angle Formed by Two Intersecting Line
If line g and line h are intersected, then the angle between them is the acute angle, α.
Notation : <(g, h) = α.
b. Angle Formed by Two Crossing Line
If line g and line h cross over each other, then the angle formed can be determined as follows:
a) Let any point A on line g. Rasa
b) Make line h' through A and parallel to line h.

26. c) The angle size formed by line g and line h' is the angle size of the line g and line h and notated as
<(g, h) ≡ <(g, h') = α.
or
(1) Make line g' parallel to line g.
(2) Make line h' intersecting g' and parallel to h.
(3) The angle size formed by line g' and line h' is the angle size of line g and line h and notated as
<(g, h) ≡ <(g', h') = α.

27. Example:
Given a cube ABCD. EFGH with edge length a cm. Find the angle size between:
a. Line AH and line BF
b. Line DF and line BG
c. Line DE and line HF
Answer:
a. Angle between line AH and line BF is <(AH, AE) = 𝛼1 = 45°, because
BF is parallel to AE.
b. Angle between line DE and line BG is <(CF, BG) = 𝛼2 = 90°, because
CF is parallel to DE and CF is intersecting perpendicular to BG.
c. Angle between line DE and line BG is <(DE, DB) = 𝛼3 = 60°, because ∆𝐵𝐸𝐷 is equilateral triangle and
HF is parallel to DB.

28. 2. Angle Formed by a Line and a Plane
Suppose you are given a line l and plane V. To find the angle size between l and V can follow these
ways. Lengthen line l until in intersects plane V on point P. Then, by projecting line l onto plane V, get line
l’. Angle between line l and plane V is the angle formed by lengthened line l and line l’, which is 𝛼.

29. Example:
Given a cube ABCD. EFGH with edge length 10 cm.
a. Draw an angle between line AG and plane ABCD
b. Measure the angle size.
Answer:
a. Projection of line AG onto plane ABCD is line AC. So, the angle
between line AG and plane ABCD is <GAC = 𝛼.
b. See that CG = 10 cm and AC = 10√2 𝑐𝑚 because AC is the
diagonal of cube’s face. See that GAC has a right angle on C,
then
tan 𝛼 =
𝐶𝐺
𝐴𝐶
=
10
10√2
=
1
2
√2 𝑎𝑛𝑑 𝛼 = 35,3°
So, the angle size between line AG and plane ABCD is 𝛼 = 35,3°.
3. Angle Formed by Two Planes
a. Angle Between Two Parallel Planes and One on the Other
If plane V is on plane W or plane V is parallel to plane W, then the angle between these planes is
<(V, W) = 0°.

30. b. Angle Between Two Intersecting Planes
If plane V intersects plane W on line (V, W), then the angle between planes V and W can be
determined as follows.
a) Let any point P on line (V, W).
b) Make line g on plane V through P and perpendicular to line (V, W).
c) Make line h on plane W through P and perpendicular to line (V, W).
d) Angle 𝛼 is formed by line g and line h.
See that the angle formed by two intersecting planes can be represented by the angle formed by line g
and line h, which is 𝛼. In other words, angle between two intersecting planes is defined by the angle
formed by two lines on both planes where the lines are perpendicular to the intersection line of planes.

31. Example:
Given a cube ABCD.EFGH with edge length a units. Sketch and calculate the size of angle between plane
BDE and plane BDG.
Answer:
See the following figure. An angle between plane BDE and
plane BDG is 𝛼. See ∆EPA has a right angle on A, thus
𝑃𝐸 = √𝐴𝑃2 + 𝐴𝐸2
= √(
1
2
𝑎√2)
2
+ 𝑎2
= √
1
2
𝑎2 + 𝑎2 = √
3
2
𝑎2 =
1
2
𝑎√6
Bacuse ∆GCP ≡ ∆EPA, then PG = PE =
1
2
𝑎√6
See ∆EG.
From Cosine Rule, resulting
cos 𝛼 =
𝑃𝐸2
+ 𝑃𝐺2
− 𝐸𝐺2
2 . 𝑃𝐸 . 𝑃𝐺

32. =
(
1
2
𝑎√6)
2
+(
1
2
𝑎√6)
2
−(𝑎√2)
2
2 .
1
2
𝑎√6 .
1
2
𝑎√6
=
𝑎2
3𝑎2 =
1
3
𝛼 = 70,53°
So, the size angle between plane BDE and plane BDG is 𝛼 = 70,53°