similar shapes - ähnliche Figuren
introduction
what is similarity?
task 1
What shapes are similar?
TODO - page with similar shapes.
What can you say about the angles when they are similar?
What can you say about the ratios of matching sides to each other?
task 2 - a tree and its shadow.
How can we determine the height of a tree with just a measuring tape?
similarity - Ähnlichkeit
task enlarge a drawing
Let’s say you want to enlarge a drawing that is on square paper.
make your own drawing.
stretch factor (enlargement factor) - Ähnlichkeitsfaktor
If two shapes A and B are similar, we write $A \sim B$
.
$\frac{1}{4}$
, 25 %, 1 : 4 and 0.25 mean the same scale factor
enlargement / reduction - zentrische Streckung
experiment
materials
- rubber band (knot - half way)
- pin
- wooden board (or the like to stick the pin in)
procedure
make sure the knot follows the curve you want to enlarge. Keep the pin and paper fixed.
video
here is a link to a video demonstration
Extension: - What happens if the knot is not halfway?
theory
- scaling - zentrische Streckung
- centre of enlargement - Streckzentrum Z
- enlargement Vergrößerung
- reduction Verkleinerung
Note: always measure from point Z.
The shapes are similar. Die Figuren sind ähnlich.
Strahlensätze
1. Strahlensatz
$${\frac{a_1}{a_2} = \frac{b_1}{b_2}}$$
2. Strahlensatz
$${\frac{a_1}{a_2} = \frac{c_1}{c_2}}$$ and $${\frac{b_1}{b_2} = \frac{c_1}{c_2}}$$
ratios of lengths - Verhältnisse der Längen
$${l_{image} = l_{original} \cdot k}$$
what about the area of shapes?
example - rectangle - ${A = a \cdot b}$
image: ${A' = (ak) \cdot (bk) = a \cdot b \cdot k \cdot k = ab \cdot k^2 = A \cdot k^2}$