toan Tieng Anh " The Distance"

For all M on the line a, we can see that OH OM OH is less than or equal to OM.. Hence, d is the least length from O to any point on the line... The distance from a point to a plane:• Giv

THE DISTANCE

Introducing terms used in the

lesson

• perpendicular

• parallel

• projector

• right triangle

• square

• distance

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I The distance from a point to a line :

• Give the line a and one point O.

a

O

We get the distance from O to the line a is the

length of OH The notation is: d(O,a) = OH = d.

For all M on the line (a), we can see that OH OM (OH is less than or equal to OM) Hence, d is the least length from O to any point on the line.

• Let H is the projector of O on the line a.

≤

II The distance from a point to a plane:

• Give the plane (P) and one point O, H is the projector of O

on the plane.

• We get the distance from O to (P) is the length of OH

The notation is d(O;(P)) = d = OH.

• For all M on the plane, we can see that OH OM (OH

is less than or equal OM).

• Hence, d is the least length from O to any point on

the plane.

≤

III The distance from a line to a plane that is parallel to that line.

• Let d is the distance from the line a to the plane (P) that is parallel to a

• We define that d is equal to the distance from any

point on the line a to the plane (P)

The notation is: d(a,(P)) = MH

• Let S.ABCD be pyramid with ABCD is a

square edge a, SA is perpendicular to the

plane (ABCD) and the length of the line

SA =

a) Calculate the distance from A to plane (SCD).

b) Calculate the distance between the

straight line CD and mp (SAB).

a 2

1) Let H is the projector of A on the line SD.

• We see: SD ⊥ AH (1)

• Moreover CD ⊥ AD, CD ⊥ AD so CD ⊥ (SAD) imply CD ⊥ AH (2)

• From (1), (2), We get AH ⊥ (SCD)

• Hence AH = d(A,(SCD))

2) We calculated AH = 2) We see CD parallel to (SAB) so d(CD, (SAB)) = d(D,(SAB)) = AD = a

A

D

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• Exercise 1: Let ABC.A ′ B ′ C ′ be prismatic with AA ⊥ (ABC), AA ′ = a, right triangle

ABC at A where BC = 2a, AB = a

Calculate:

a) The distance from line AA ′ to the plane (BCC ′ B ′ ).

b) The distance from A to (A ′ BC)

c) Prove that AB ⊥ (ACC’A’) and the

distance from point A ′ to the plane (ABC’)