Three perpendiculars and perpendicular planes
Answers are for parents and teachers.
1
DA ⊥ (ABC), DA = 9 and the altitude from A to BC is AO = 12. Find the distance from D to BC.
15
2
In triangle ABC, AB = AC = 17 and BC = 16; DA ⊥ (ABC), DA = 15. Find the distance from D to line BC.
AO = √(17² − 8²) = 15 (O is the midpoint of BC); DO ⊥ BC, DO = √(15² + 15²) = 15√2.
3
In a cube, how many degrees is the angle between planes ACC₁A₁ and ABCD?
90°
4
A regular quadrilateral pyramid has base side 4 and apothem 4. Find the angle between a lateral face and the base.
r = 2, cos φ = 2/4 = 1/2, φ = 60°.
5
Why does the theorem of three perpendiculars help find the distance from a point to a line?
It shows that the segment from D to the foot of the perpendicular from A to the line is itself perpendicular to the line, so the distance is DO = √(DA² + AO²).