One of the largest issues in ancient mathematics was accuracy — nobody had calculators that went out ten decimal places, and accuracy generally got worse as the numbers got larger. The famous Eratosthenes experiment, that can be found at famousscientists,org/eratosthenes/, relied on the fact known to Thales and others that a beam of parallels cut by a transverse straight line determines an equal measure for the corresponding angles. Given two similar triangles, one with small measurements that can be accurately determined, and the other with large measurements, but at least one is known with accuracy, can the other two measurements be deduced? Explain and give an example.&nbsp;The similarity of triangles gives rise to trigonometry.&nbsp;How could we understand that the right triangles of trigonometry with a hypotenuse of measure 1 represent all possible right triangles? Ultimately, the similarity of triangles is the basis for proportions between sides of two triangles, and these proportions allow for the calculations of which we are speaking here. The similarity of triangles is the foundation of trigonometry.

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One of the largest issues in ancient mathematics was accuracy — nobody had calculators that went out ten decimal places, and accuracy generally got worse as the numbers got larger. The famous Eratosthenes experiment, that can be found at famousscientists,org/eratosthenes/, relied on the fact known to Thales and others that a beam of parallels cut by a transverse straight line determines an equal measure for the corresponding angles. Given two similar triangles, one with small measurements that can be accurately determined, and the other with large measurements, but at least one is known with accuracy, can the other two measurements be deduced? Explain and give an example.&amp;nbsp;The similarity of triangles gives rise to trigonometry.&amp;nbsp;How could we understand that the right triangles of trigonometry with a hypotenuse of measure 1 represent all possible right triangles? Ultimately, the similarity of triangles is the basis for proportions between sides of two triangles, and these proportions allow for the calculations of which we are speaking here. The similarity of triangles is the foundation of trigonometry.

taibidzhl · Accepted Answer

Suppose XYZ and KLM are two similar triangles.&amp;nbsp;Their sides are in proportion.&amp;nbsp;⇒XYKL=YZLM=XZKM&amp;nbsp;Suppose&amp;nbsp;△XYZ&amp;nbsp;is a smaller triangle.&amp;nbsp;Measure sides XY, YZ and XZ of&amp;nbsp;△XYZ&amp;nbsp;Suppose one of the side of a triangle&amp;nbsp;△KLM&amp;nbsp;Let KM = k&amp;nbsp;⇒XYKL=YZLM=XZk&amp;nbsp;⇒KL=k×XY&amp;nbsp;and&amp;nbsp;LM=k×YZ&amp;nbsp;k, XY and YZ are know we can determine KL and LM.&amp;nbsp;&amp;nbsp;So, yes other two measurements can be deduced.

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