Orthogonal projection and perpendicularity in practice
Lessons 51–52 · 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
52
Practice and applications: perpendicularity
Textbook, Part 2: pp. 140–143
GoalApply knowledge of perpendicularity to problems (distances, angles, areas) and to practical checks (verticality, right angles).
New words
verticality · tiklikplumb line · shoqulspace diagonal · fazoviy diagonaldistance · masofa
Explanation
In perpendicularity problems first decide by the criterion that a line is perpendicular to a plane, then find right triangles and use the Pythagorean theorem. For distances: from a point to a plane it is the perpendicular; from a point to a line it is the hypotenuse of a right triangle found with the theorem of three perpendiculars. For angles: the angle between a line and a plane is the angle between the line and its projection, and the angle between planes is the linear angle. Practical checks rest on the same ideas: we check a post’s verticality with a plumb line from two different directions, because if the post is the line of intersection of two planes, both planes contain the vertical plumb line; we check a room corner with a 3-4-5 triangle: sides 3, 4, 5 give a right angle; a door stays perpendicular to the floor because its axis (the hinge line) is perpendicular to the floor.
Worked examples
In pyramid SABCD, ABCD is a square of side 6, SA ⊥ (ABCD) and SA = 6. The angle of SB with the base: the projection is AB and SA = AB, so the angle is 45°. BC ⊥ AB and AB is the projection of SB, so BC ⊥ SB; triangle SBC is right-angled: SC = √(72 + 36) = 6√3.
To check that a post is perpendicular to the ground we compare it with a plumb line from two points: from each point the post and the string appear in one plane. The post lies on the line where the two planes meet, and the plumb line is in each of those planes; so post and plumb line are parallel — the post is vertical.
Class activity
Try the 3-4-5 method on a notebook or room corner: measure 3 units along one wall and 4 along the other from the corner; if the distance between the ends is 5 units, the corner is right-angled. (Use a tape measure with an adult in a room.)
Practice
1
Why do we check a post with a plumb line from two directions?
Parallelism seen in one plane is not enough; checked in two intersecting planes, the post is parallel to their line of intersection — vertical.
2
SA ⊥ (ABCD), ABCD is a square of side 6 and SA = 6√3. How many degrees is the angle between SB and the base?
The projection is AB; tan φ = SA/AB = √3, φ = 60°.
3
What is the length of the longest straight rod that fits in a box with dimensions 1, 4, 8?
9
4
Why does an open door remain a plane perpendicular to the floor?
The door contains the hinge axis, which is perpendicular to the floor; a plane through a line perpendicular to a plane is perpendicular to it (criterion).