Parallel lines project to parallel lines
Opposite sides stay side by side.
A card in the sunlight
Hold a rectangular card in the sunlight. Its shadow on the ground has four sides. Now turn the card this way and that.
What can the shadow look like?
What this lesson covers
The idea
The projection of a pair of parallel lines always remains parallel, so a parallelogram, however it is oriented, projects to a parallelogram.
A card in the sunlight
Hold a rectangular card in the sunlight. Its shadow on the ground has four sides. Now turn the card this way and that.
What can the shadow look like?
- Any four-sided shape
- Always a shape with opposite sides parallel
- Sometimes a triangle
Turn and tip the card
A rectangular card hovers above the floor. Each corner drops straight down, and the dark shape is its projection. The toy measures the angle between the long sides and between the short sides. The dotted lines are the sides, made longer.
Parallel stays parallel
The projection of a pair of parallel lines always remains parallel. So a parallelogram, however it is turned, has a parallelogram as its projection.
Lengths and corner angles do change: a rectangle gave a slanted parallelogram. Only the parallel sides stay parallel.
A lamp close to the card gave a distorted shadow. A projection, or the shadow in the Sun, keeps parallel lines parallel.
Notes
The projection of a pair of parallel lines always remains parallel, so a parallelogram, however it is oriented, projects to a parallelogram.
Check yourself
A rectangular card is projected on the floor. Which of these can not be its projection?
The projection of a parallelogram on the floor has one angle of 50°. What is the angle next to it?
Answer: 130 °
The projection of a parallelogram is a parallelogram, and in a parallelogram two neighbouring angles add up to 180°. 180° − 50° = 130°.
Which statement is true for the projection of a rectangular card on the floor?
- Shape A. A is a rectangle. A card held flat, or tipped about its side, gives a rectangle.
- Shape B. B is a slanted parallelogram. A card turned and tipped gives one, as you saw.
- Shape C — correct. Yes! C has only one pair of parallel sides. A projection keeps both pairs of opposite sides parallel.
- Its opposite sides are always parallel. — correct. Yes! A pair of parallel lines always projects to a pair of parallel lines.
- Its corners are always right angles.. The corners can change. When the card was turned and tipped, the corner angle was not 90°.
- Its sides are always as long as the sides of the card.. A tipped card gives a shorter projection. Lengths can change; parallel lines stay parallel.