THE DISTANCE Introducing terms used in the lesson • perpendicular • parallel • projector • right triangle • square • distance A D B C S K I. The distance from a point to a line: • Give the line a and one point O. a O H M 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 Example • 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 B C S K • 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’). . 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. point to a line: • Give the line a and one point O. a O H M 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. 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