diff --git a/translations/de/transformations/content.md b/translations/de/transformations/content.md index 0442ec7df..e214e10a7 100644 --- a/translations/de/transformations/content.md +++ b/translations/de/transformations/content.md @@ -4,6 +4,7 @@ > id: intro > section: introduction +> description: Symmetrie kann man überall in der Natur sehen - aber sie liegt auch völlig unsichtbaren Naturgesetzen zugrunde. Die Mathematik kann erklären, warum das so ist. > color: "#2274E8" > level: Intermediate > next: triangles @@ -13,11 +14,12 @@ wurden von Mathematikern "erfunden". Symmetrie hingegen ist überall um uns herum. Fast alle Pflanzen, Tiere und sogar wir Menschen sind symmetrisch. ::: column(width=200) - x-img(src="images/butterfly.jpg" width=200 height=200 lightbox) + x-img(src="images/butterfly.jpg" width=200 height=200 lightbox alt="Schmetterling") ::: column(width=200) - x-img(src="images/lion.jpg" width=200 height=200 lightbox) + x-img(src="images/lion.jpg" width=200 height=200 lightbox alt="Löwe") ::: column(width=200) - x-img(src="images/starfish.jpg" width=200 height=200 lightbox) + x-img(src="images/starfish.jpg" width=200 height=200 lightbox alt="Seestern") + ::: Im Laufe der Zeit haben wir die Symmetrie der Natur in Kunst, Architektur, Technologie @@ -684,8 +686,8 @@ Fallen dir noch andere Beispiele für Palindrome ein? Wenn wir Leerzeichen und Interpunktion ignorieren, sind auch die kurzen Sätze unten spiegelsymmetrisch. Fallen dir eigene Beispiele ein? -{.text-center} Ein Esel lese nie. -Sei mein, [[nie]] fies - sei fein, nie mies. +{.text-center} Ein Esel lese nie.
+Sei mein, [[nie]] fies - sei fein, nie mies.
Trug Tim eine so helle Hose nie [[mit]] Gurt? {.reveal(when="blank-0 blank-1")} Aber Palindrome machen nicht nur Spaß, sie haben @@ -1284,5 +1286,209 @@ Bild [[größer|kleiner]] als das Original. > section: similarity > sectionStatus: dev +> id: similarity -TODO +::: column.grow +Bei starren Transformationen ist das Bild immer [[kongruent|größer|kleiner]] als das Original - für Streckungen gilt das aber [[nicht mehr|auch]]. Stattdessen sagen wir, dass zwei Formen [__ähnlich__](gloss:similar) sind. Sie haben im zwar die gleiche +Form, aber nicht unbedingt die gleiche Größe. + +Das Symbol für Ähnlichkeit ist `∼` (ähnlich wie das Symbol für Kongruenz, also `≅`). In diesem Beispiel würden wir `A ∼ A'` schreiben. + +::: column(width=240) +{.todo} COMING SOON – Illustration +::: + +--- +> id: perspective + +### Perspective Drawings + +You might have noticed that these dilations with the connecting rays almost look +like __perspective drawings__. The center of dilation is called the __vanishing +point__, because it looks like this is where everything is “vanishing in the +distance”. + +Find the vanishing point in the figure below: + +{.todo} COMING SOON – Interactive + +Now can you draw another house that matches the existing ones? + +--- +> id: similar-polygons + +### Similar Polygons + +Similarity can tell us a lot about shapes. For example, [circles](gloss:circle), +[squares](gloss:square) and [equilateral triangles](gloss:equilateral-triangle) +are [[always|sometimes|never]] similar. They might have different sizes, but +always the same general shape. + +::: column.grow +The two quadrilaterals on the right are similar. Our first important observation +is that in similar polygons, all the matching pairs of angles are +[congruent](gloss:congruent-angles). This means that + +{.text-center} [_{.m-red}∡ABC_ ≅ _{.m-red}∡A'B'C'_](target:a)_{.space}_ +[_{.m-blue}∡BCD_ ≅ _{.m-blue}∡B'C'D'_](target:b) +[_{.m-green}∡CDE_ ≅ _{.m-green}∡C'D'E'_](target:c)_{.space}_ +[_{.m-yellow}∡DEA_ ≅ _{.m-yellow}∡D'E'A'_](target:d) + +The second important fact is that in similar polygons, all sides are scaled +__proportionally__ by the scale factor of the corresponding dilation. If the +scale factor is ${k}{k|1.5|0.5,2,0.1}, then + +{.text-center} `abs(AB) ×` ${k} `= abs(A'B')`_{.space}_`abs(BC) ×` ${k} `= abs(B'C')` +`abs(CD) ×` ${k} `= abs(C'D')`_{.space}_`abs(DE) ×` ${k} `= abs(D'E')` + +We can instead rearrange these equations and eliminate the scale factor +entirely: + +{.text-center} `abs(AB)/abs(A'B') = abs(BC)/abs(B'C') = abs(AB)/abs(A'B') = abs(AB)/abs(A'B')` + + // This proportional relationship is true not just for the sides of the + // polygon, but also for properties like diagonals. + +We can use this to solve real life problems that involve similar polygons – for +example finding the length of missing sides, if we know some of the other sides. +In the following section you will see a few examples. +::: column(width=240) + + x-geopad.sticky(width=240 height=360): svg + - var x = ['a', 'b', 'c', 'd'] + - var initial = {a:[50,70], b:[160,50], c:[200,110], d:[150,160]} + - var next = {a:'b', b:'c', c:'d', d:'a'} + - var prev = {a:'d', b:'a', c:'b', d:'c'} + - var classes = {a:'red', b:'blue', c:'green', d:'yellow'} + each l in x + circle(name=l x=`point(${initial[l][0]},${initial[l][1]})` r=4 target=l) + path(x=`angle(${prev[l]},${l},${next[l]})` target=l class=classes[l]) + path(x=`segment(${l},${next[l]})` target=`${l} ${next[l]}`) + circle(name=l+'1' r=4 x=`${l}.subtract({x:120,y:90}).scale(k).rotate(3).add({x:120,y:270})` target=l) + path(x=`angle(${prev[l]}1,${l}1,${next[l]}1)` target=l class=classes[l]) + path(x=`segment(${l}1,${next[l]}1)` target=`${l} ${next[l]}`) +::: + +--- +> id: similar-triangles + +### Similar Triangles + +The concept of similarity is particularly powerful with triangles. We already +know that the corresponding internal angles in similar polygons are equal. + +For triangles, the opposite is also true: this means that if you have two +triangles with the same three angle sizes, then the triangles must be similar. + +And it gets even better! We know that the internal angles in a triangle always +add up to [[180]]°. This means that if we know two angles in a triangle, we can +always work out the third one. + +For similarity, this means that we also just need to check _two angles_ to +determine if triangles are similar. If two triangles have two angles of the same +size, then the third angle must also be the same in both. + +This result is sometimes called the [__AA Similarity Condition__](gloss:triangle-aa) +for triangles. (The two _As_ stand for the two _angles_ we compare.) + +::: .theorem +If two angles in one triangle are congruent to two angles in another triangle, +the two triangles are similar. +::: + +--- +> id: similar-triangles-1 + +Let’s have a look at a few examples where this is useful: + +::: column(width=320) +{.todo} COMING SOON – Animation + +::: column.grow +Here you can see the image of a large lighthouse. Together with a friend, you +want to measure the height of the lighthouse, but unfortunately we cannot climb +to the top. + +It turns out that, very well hidden, the diagram contains two similar triangles: +one is formed by the lighthouse and its shadow, and one is formed by your friend +and her shadow. + +Both triangles have one right angle at the bottom. The sun rays are parallel, +which means that the other two angles at the bottom are corresponding angles, +and also equal. By the AA condition for triangles, these two must be similar. + +We can easily measure the length of the shadows, and we also know the height of +your friend. Now we can use the proportionality of sides in similar triangles +to find the height of the lighthouse: + +{.todo} COMING SOON – Equation + +Therefore the lighthouse is 1.5m tall. +::: + +--- +> id: similar-triangles-2 + +::: column(width=320) +{.todo} COMING SOON – Animation +::: column.grow +We can use the same technique to measure distances on the ground. Here we want +to find the width of a large river. There is a big tree on one side of the +river, and I’ve got a stick that is one meter long. + +Try drawing another two similar triangles in this diagram. + +You can mark the point along the side of the river, that lies directly on the +line of sight from the end of the stick to the tree. Then we can measure the +distances to the stick, and to the point directly opposite the tree. + +Once again, these two triangles are similar because of the AA condition. They +both have a right angle, and on pair of opposite angles. + +According to the proportionality rule, this means that + +{.todo} COMING SOON – Equation + +Therefore the width of the river is 45 meters. +::: + +--- + +### Similarity on Rays + +Theorem: If a ray bisects an angle of a triangle, then it divides the +opposite side into segments that are proportional to the lengths of the +other two sides. + +We can extend this theorem to a situation outside of triangles where we +have multiple parallel lines cut by transverals. + +Theorem: If three or more parallel lines are cut by two transversals, then they +divide the transversals proportionally. + +Think about a midsegment of a triangle. A midsegment is parallel to one side of +a triangle and divides the other two sides into congruent halves. The midsegment +divides those two sides proportionally. + +Triangle Proportionality Theorem: If a line parallel to one side of a triangle +intersects the other two sides, then it divides those sides proportionally. + +Triangle Proportionality Theorem Converse: If a line divides two sides of a +triangle proportionally, then it is parallel to the third side. + +--- + +### Self Similarity + +There are some curious mathematical shapes that are similar to a smaller part +_of themselves_. An example is the __Sierpinksi Triangle__: the entire triangle +is similar to any one of the smaller triangles it consists on. You could zoom +in and infinitely many smaller and smaller triangles. + +Shapes with this property are called __Fractals__. They have some surprising +and truly XXX properties, which you will learn about more in the future. + +--- + +Triangles are not just useful for measuring distances. In the next course we +will learn a lot more about triangles and their properties.