Perpendicular lines and planes, obliques, distance
Lessons 47–48 · 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
47
Perpendicular lines and planes in space
Textbook, Part 2: pp. 119–123
GoalKnow perpendicular lines in space, a line perpendicular to a plane, its criterion and properties, and the formula for the diagonal of a rectangular parallelepiped.
New words
line perpendicular to a plane · tekislikka perpendikular chiziqcriterion for perpendicularity · perpendikularlik alomatirectangular parallelepiped · to‘g‘ri burchakli parallelepipeddiagonal · diagonal
Explanation
Two lines with a 90° angle between them are perpendicular; they may be intersecting or skew (a ⊥ b). A line is perpendicular to a plane if it is perpendicular to every line of that plane. Criterion: if a line is perpendicular to two intersecting lines of a plane, it is perpendicular to the plane. Properties: two lines perpendicular to the same plane are parallel; a line perpendicular to one of two parallel planes is perpendicular to the other; two planes perpendicular to the same line are parallel; through every point of space there is exactly one plane perpendicular to a given line and exactly one line perpendicular to a given plane. For a rectangular parallelepiped with dimensions a, b, c the diagonal d satisfies d² = a² + b² + c² (the generalised Pythagorean theorem).
Worked examples
A rectangular parallelepiped has dimensions 2, 6, 9. d² = 4 + 36 + 81 = 121, so d = 11.
Segment SA is perpendicular to plane ABC when SA ⊥ AB and SA ⊥ AC (AB, AC intersecting). By the criterion SA ⊥ (ABC), hence SA is perpendicular to every line of the plane, including BC: SA ⊥ BC.
Class activity
Use a corner of a room as a model: two walls and the floor give three mutually perpendicular lines (edges). Measure the dimensions with a tape measure (with an adult’s help) and compute the room’s space diagonal by the formula.
Practice
1
State the criterion for a line to be perpendicular to a plane.
If it is perpendicular to two intersecting lines of the plane.
2
A rectangular parallelepiped has dimensions 3, 4, 12. Find its diagonal.
13
3
In the rectangular parallelepiped ABCDA₁B₁C₁D₁, AB = 6, AD = 8, AA₁ = 24. Find the base diagonal BD and the space diagonal BD₁.