Euthym wrote:Here you know all the angles and a length, so you can use the sinus rule multiple times, but also Pythagoreans theorem once you know another lenght. We have a triangle ABC, with A=90°, B=10°, C=80°. Using the sinus rule we can calculate that the bottom of the left triangle is 1,76 cm. So the triangle on the right must have 2 lenghts of 10-1.76cm=8.24. Using the pythagorean theorem ce can calculate that y=11.65cm
Baltorigamist wrote:Gerardo -> For the 75deg example, 90 - 75 = 15--and a 15deg angle is easy to construct. You'd simply mark out 1/2 of the bottom edge and fold through the upper left corner such that the bottom corner touches the 1/2 mark.
Note that this gives you a 75deg angle from the top edge--not centered along the main diagonal--but you just need to bisect the 15deg angle and repeat in mirror image on the top edge.
Baltorigamist wrote:All you need to do is calculate the sine of the smaller angles (in degrees) of the triangles and multiply that by the side of the square.
For the 75deg example, (90-75)/2=7.5 degrees. Sin(7.5) is about .1305, which means the bottom left triangle has a base of .1305 times the side of the square, or 1.305cm. Subtracting that from the 10cm side length leaves us with 8.695cm for the short side(s) of the bottom right triangle. Multiply that by sqrt2, or ~1.41421, for the total hypotenuse length.
The 70deg example can be done by following the same steps (but using sin(10) instead of sin(7.5)).
Baltorigamist wrote:The base:hypotenuse ratio of the triangles is 2.1:3.65 (or about 1:1.74), so we need to divide 2.9 into that proportion. To do that, we add the two parts together and divide 2.9 by the result: 2.9/2.74=~1.058. Multiplying that by 1.74 gives the hypotenuse of the smaller triangle (a in your example) as 1.841cm. The value of b, then, is (2(1.058+2.1)), or a little less than 6.32cm.
Baltorigamist wrote:Having the desired ratio in simplest terms means that dividing 2.9 by the sum of the proportions (in this case, 2.74) will give me one side of the proportion directly
Baltorigamist wrote:I just need to multiply the result (1.058) by the rest of the proportion to find the second part of the ratio [the hypotenuse of the small right triangle on the bottom right].
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