File:Academ scale ratio tan 60deg.svg
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English: The image shows the values of tan(60°) and sin(60°). The triangle BCK is equilateral. The orthogonal projection onto (KC) transforms the midpoint A of [BK] into H. A tiling fills BCK with eight right triangles, congruent one to another, four on either side of the bisector of [BK]. One of them is HKA. The tiling fills HAC with three triangles numbered from 1 to 3, it fills ABC with four numbered from 1 to 4. A similarity of which the scale ratio is 1/2 and the angle -120° (where counterclockwise turning is positive) transforms ABC into HKA, dividing by four its area: (HA/AC)² = 1/4. A similarity of which the center is H and the angle +90° transforms HKA into HAC, multiplying by three its area. Thus the scale ratio of this similarity is: tan(60°) = HC/HA = HA/HK = AC/KA = √3. The composition of the two similarities is the similarity of which the center is C, the angle -30°, and the scale ratio sin(60°). It transforms two sides of ABC drawn in red into two sides of HAC drawn in green. The two corresponding sides of HKA are dashed in blue.
Français : L’image met en évidence les valeurs de tan(60°) et sin(60°). Le triangle BCK est équilatéral. Le point H est le projeté orthogonal sur (KC) du milieu A de [BK]. Huit triangles rectangles tous isométriques entre eux constituent un pavage de BCK, quatre de part et d’autre de la médiatrice de [BK]. L'un d’eux est HKA. Le pavage remplit HAC de trois triangles numérotés de 1 à 3, et ABC de quatre triangles numérotés de 1 à 4. Une similitude de rapport 1/2 et d’angle -120° (avec le sens de rotation anti-horaire positif) transforme ABC en HKA en divisant son aire par 4 : (HA/AC)² = 1/4. Une similitude de centre H et d’angle +90° transforme HKA en HAC en multipliant son aire par trois. Le rapport de cette similitude est donc : tan(60°) = HC/HA = HA/HK = AC/KA = √3. La composée des deux similitudes est la similitude de centre C, d’angle -30°, et de rapport sin(60°). Elle transforme deux côtés de ABC tracés en rouge en deux côtés de HAC tracés en vert. Les deux côtés correspondants de HKA sont en pointillé bleu. |
| Date | |
| Source | Own work |
| Author | Yves Baelde |
| Other versions |
Published in France in “Souci d’exactitude” (Yves Baelde), Bulletin de l’APMEP n° 401, december 1995 (p.877). Other version: File:Academ_scale_ratio_cos_30deg.svg |
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 09:38, 23 May 2010 | | 450 × 360 (4 KB) | Baelde (talk | contribs) | {{Information |Description={{en|1=The image shows the values of ''tan''(60°) and ''sin''(60°). The triangle BCK is equilateral. The orthogonal projection onto (KC) transforms the midpoint A of [BK] into H. A tiling fills BCK with eight right trian |
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