On a flat sheet of paper, two lines either meet or are parallel.
In space there is a third possibility, skew lines that neither meet nor are parallel, and lines and planes, or two planes, can also be parallel, meet or be perpendicular.
How far two planes open apart is measured by the dihedral angle; think of the angle of an open book.
Turn the view to see the x-, y- and z-axes and the point P in three dimensions, and change the dihedral angle to see the angle between the two rays perpendicular to the edge change with it.
Coordinate Axes in Space
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The x-, y- and z-axes with the point P(2, 2, 3) and the foot H of the perpendicular to the xy-plane
30°
👀 Three-dimensional space
①Add a z-axis perpendicular to both the x-axis and the y-axis of the plane, and you have space
②The three axes are mutually perpendicular, and the axis pointing up is taken as the z-axis
③Dropping a perpendicular from P(2, 2, 3) to the xy-plane gives the foot H(2, 2, 0), and P's z-coordinate 3 is the length of PH
④Every point in space is given by three coordinates (x, y, z)
Positions of Lines and Planes
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Two parallel planes · two intersecting planes · a line perpendicular to a plane
∥
📐 Relative positions
①∥ : the two planes are parallel and never meet
②∩ : the two planes meet in a single line, their line of intersection
③ℓ ⊥ α : a line perpendicular to a plane is perpendicular to every line in that plane
④Besides meeting and being parallel, two lines in space can be skew: neither meeting nor parallel
Dihedral Angle
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Two half-planes α and β sharing an edge, and the dihedral angle θ
60°
📖 Measuring a dihedral angle
①The two half-planes share an edge (green)
②From a point on the edge, draw in each half-plane a ray perpendicular to the edge
③The angle θ between these two rays is the dihedral angle
④The angle between the two covers of an open book is a dihedral angle
Perpendicularity and Parallelism in Space
Theorem of three perpendiculars
If PH ⊥ α, then (PO ⊥ ℓ ⟺ HO ⊥ ℓ). P is a point off the plane, H the foot of the perpendicular, O a point on the line ℓ in the plane
Decide whether a point off a plane and a line in the plane are perpendicular by using the foot of the perpendicular
A line parallel to a plane
ℓ ∥ α ⟺ ℓ ∥ m (for some line m ⊂ α)
A line not in a plane is parallel to the plane if it is parallel to some line in the plane
🔍 Key theorems
①A line perpendicular to a plane is perpendicular to every line in the plane
②A plane containing a line perpendicular to another plane is perpendicular to that plane
③The theorem of three perpendiculars decides perpendicularity by way of the foot of a perpendicular
④Drawing each theorem and marking the foot of the perpendicular makes the proofs easy to follow
Worked Examples
Example 1: theorem of three perpendiculars
A line ℓ lies in plane α, and P is a point not in α. Let H be the foot of the perpendicular from P to α, and O the foot of the perpendicular from H to ℓ. If PH = 3 and HO = 4, find PO.
1
By the theorem of three perpendiculars, PO ⊥ ℓ, and triangle PHO has its right angle at H.
PH ⊥ α, HO ⊥ ℓ ⇒ PO ⊥ ℓ
2
Use the Pythagorean theorem in right triangle PHO to find the hypotenuse PO.
PO = √(PH² + HO²) = √(9 + 16) = √25
▸ PO = 5
Since PH is perpendicular to plane α, it is also perpendicular to the segment HO in α, which makes PHO a right triangle.
Example 2: projected area
A figure of area 12 lies in a plane β that makes a 60° angle with plane α. Find the area of its projection onto α.
1
The projected area is the original area times the cosine of the angle between the planes.
S′ = S · cos θ
2
Substitute S = 12 and θ = 60°.
S′ = 12 · cos 60° = 12 · (1/2)
▸ S′ = 6
The larger the angle, the smaller the cosine and the projection. At 90° the projected area is 0.
Summary
Area of a projection
S′ = S cos θ
The area of a plane figure projected onto another plane is its original area times the cosine of the angle between the planes
2021 KICE mock exam Math (Geometry) type, adapted
A square of side 4 makes a 45° angle with plane α. What is the area of its projection onto α?
①8
②8√2
③16
④4√2
⑤4
▸ ② 8√2
1
The area of the square is 4 × 4 = 16.
2
Substitute S = 16 and θ = 45° into the projected-area formula.
S′ = 16 · cos 45° = 16 · (√2/2) = 8√2
🎯 Exam Points
①Tell apart the possible positions of lines and planes in space
②State the conditions and conclusion of the theorem of three perpendiculars exactly
③Measure a dihedral angle with two rays perpendicular to the edge
④Projected area = original area × cos θ
⑤Be able to find the distance between two skew lines