Lessons 49–50 · 2 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 10, Parts 1 and 2, 1st edition. EXTREMUM PRESS, Tashkent, 2017
49
The theorem of three perpendiculars
Textbook, Part 2: pp. 128–131
GoalKnow the theorem of three perpendiculars and its converse and the angle between a line and a plane, and use them in problems.
New words
theorem of three perpendiculars · uch perpendikularlar teoremasiprojection · proyeksiyaangle between a line and a plane · to‘g‘ri chiziq va tekislik orasidagi burchakconverse theorem · teskari teorema
Explanation
Let AB be perpendicular to plane α, AC an oblique and BC its projection. Theorem of three perpendiculars: if a line c in α through the foot C of the oblique is perpendicular to the projection BC, then it is also perpendicular to the oblique AC. The converse: if c is perpendicular to the oblique it is perpendicular to the projection. The name comes from the three perpendicularities: AB ⊥ α, c ⊥ BC and c ⊥ AC. The theorem is often used to find the distance from a point to a line: drop DA ⊥ plane from D, then from A draw AO ⊥ l to the line l in the plane; then DO ⊥ l and DO² = DA² + AO². The angle between a line and a plane is the angle between the line and its projection onto the plane; it is 90° for a perpendicular and 0° for a parallel line.
Worked examples
In triangle ABC, AB = AC = 10 and BC = 12. The altitude AO from A to BC bisects the base: AO = √(10² − 6²) = 8. DA ⊥ (ABC), DA = 6. By the theorem of three perpendiculars DO ⊥ BC; DO = √(6² + 8²) = 10 is the distance from D to BC.
What angle does the diagonal AB₁ of a cube make with plane ABCD? B₁B ⊥ ABCD and the projection is AB. Triangle B₁BA is right isosceles (BB₁ = AB), so the angle is 45°.
Class activity
Check the verticality of a post or a stick with a plumb line (a light weight on a string, kept low and safe): hold the string next to the stick and look from two different places. Discuss why checking from two directions not in one plane is needed.
Practice
1
State the theorem of three perpendiculars.
If a line in the plane is perpendicular to the projection of an oblique, it is perpendicular to the oblique itself.
2
DA ⊥ (ABC), DA = 8 and the altitude from A to BC is AO = 6. What is the distance from D to BC?
10
3
How many degrees is the angle between AB₁ and plane ABCD in a cube?
45°
4
Why is the angle between a line perpendicular to a plane and the plane taken as 90°?
The projection of a perpendicular is a point, so the definition loses meaning; by convention it is 90°, matching the 90° it makes with every line of the plane.