1. The Distance of Point to
Point
B
A
1

2. Example
Given :
cube ABCD.EFGH
With the length of
H P G edge is a cm.
E F
a cm
Determine the distance
of :
D C
1. point A to C,
A a cm 2. point A to G,
a cm B
3. point A to the middle
of plane EFGH
2

3. Solution :
Look at the triangle
ABC that is
the right triangle in B,
H G
then :
E F AC = AB 2  BC 2
a cm = a2  a2
D C
= 2a 2
a cm
A a cm B =a 2
So, side diagonal AC = a 2 cm
3

4. Distance AG = ?
Look at the
tiangle ACG that is the
H right triangle in C, then :
G
E F AG = AC 2  CG 2
a cm = ( a 2 )2  a 2
D C
= 2a  a
2 2
a cm
A a cm B = 3a 2 = a 3
So, space diagonal AG = a 3 cm
4

5. Distance of AP = ?
Look at the
triangle AEP that is the right
H P G triangle in E, then :
E F
AP = AE 2  EP 2
= a  2 a 2 
2
2 1
D C a2  2 a2
1
A
=
a cm B
3 2
= 2 a = 2a 6
1
So, the distance of A to P = 2 a 6
1
cm 5

6. Projection of Point to Line
From point P, we can make
P
line m line k.
Line m intersect k in Q,
m
point Q is
k the projection
Q from P to line k
6

7. Example
H G
E Given:
F
cube ABCD.EFGH
Determine the
projection of point A
D T C to line :
A B a. BC b.BD
c. ET
(T is side diagonal
on plane ABCD).
7

8. Solution :
H G to
Projection of point A
E F
a. BC is point B
A’ (AB  BC)
D T C
A B b. BD is point T
(AC  BD)
c. ET is point A’
(AC  ET)
8

9. Example
H G
E F Given cube
6 cm
ABCD.EFGH
D C With the length of
A 6 cm B
edge is 6 cm.
The distance of
point B to line AG
is….
9

10. E
H G Solution
F
P Distance of B to AG=
D C Distance of B to P
A 6 cm B (BPAG)
G BG(side diagonal) =
6√2 cm
6√2 AG(space diagonal)
P ?
= 6√3 cm
A 6 B
10

11. G Look triangle ABG
Sin A = AG = AB
BG BP
6√2
P
? 6 2 BP
6 3
= 6
A 6 B
2
( 6 2 )( 6 ) 3 6 6
BP = x 
6 3 3 3
BP = 2√6
So, distance of B to AG = 2√6
cm
11

12. The other way by using the area of triangle
ABG :
1 1
𝐴𝐵. 𝐵𝐺 = 𝐴𝐺. 𝑃𝐵
2 2
6 . 6 2 = 6 3 . 𝑃𝐵
36 2 = 6 3 . 𝑃𝐵
36 2
𝑃𝐵 = = 2 6 𝑐𝑚
6 3

13. Exercise
T Given T.ABCD that
the base is square.
The length of base
edge is 12 cm, and
the length of TA is
D C 12√2 cm. The
distance of A
12 cm
A B to TC is….
13

14. 1. Given that a regular pyramid T.ABCD has the
base AB = 4 cm, and TA = 6 cm. Determine
the distance of :
a. Point T to line AB
b. Point T to line AC