1

The Distance of Point to Point

A

B

2

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

A B

C D

H

E F

G

a cm

a cm

a cm

P

3

Solution : Look at the triangle ABC that is the right triangle in B, then :

AC =

=

=

=

So, side diagonal AC = cm

A B

C D

H

E F

G

a cm

a cm

a cm

22BCAB

22aa

2a2

2a

2a

4

Distance AG = ? Look at the

tiangle ACG that is the

right triangle in C, then :

AG =

=

=

= =

So, space diagonal AG = cm

A B

C D

H

E F

G

a cm

a cm

a cm

22CGAC

22a)2a(

2a3 3a

3a

22aa2

5

A B

C D

H

E F

G

a cm

P

Distance of AP = ? Look at the

triangle AEP that is the right

triangle in E, then :

AP =

=

=

= =

So, the distance of A to P = cm

22EPAE

2212

2aa

2

212

aa

2

23 a 6a

21

6a21

6

Projection of Point to Line

From point P, we can make line m line k.

Line m intersect k in Q,

point Q is

the projection

from P to line k

P

Q

k

m

7

Example

Given: cube ABCD.EFGH Determine the projection of point A to line : a. BC b.BD c. ET (T is side diagonal

on plane ABCD).

A B

C D

H

E F

G

T

8

Solution : Projection of point A to

a. BC is point B

b. BD is point

c. ET is point

A B

C D

H

E F

G

T

T

A’

A’ (AC ET)

(AB BC)

(AC BD)

9

Example Given cube ABCD.EFGH With the length of edge is 6 cm. The distance of point B to line AG is….

A B

C D

H

E F

G

6 cm

6 cm

10

Solution Distance of B to AG= Distance of B to P (BPAG) BG(side diagonal) = 6√2 cm AG(space diagonal) = 6√3 cm

A B

C D

H

E F

G

6 cm

P

A B

G

P

6

6√2

?

11

Look triangle ABG Sin A = = = BP = BP = 2√6

A B

G

P

6

6√2 AG

BG

AB

BP

36

26

6

BP

36

)6)(26(

?

So, distance of B to AG = 2√6

cm

3

66

3

3x

2

The other way by using the area of triangle

ABG : 1

2 𝐴𝐵. 𝐵𝐺 =

1

2 𝐴𝐺. 𝑃𝐵

6 . 6 2 = 6 3 . 𝑃𝐵

36 2 = 6 3 . 𝑃𝐵

𝑃𝐵 = 36 2

6 3= 2 6 𝑐𝑚

13

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

T

C

A B

D

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