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 https://www.famousscientists.org/eratosthenes/, relied on the fact known to Thales and others
that a beam of parallels cut by a transverse straight line determines 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.
The similarity of triangles gives rise to trigonometry.
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.
Your Discussion should be a minimum of 250 words in length and not more than 750 words.
_________________________________________________________________________________
Given two similar triangles
Let us say ∆ ABC ≈ ∆ PQR
PIC1
Now ABC is a small triangle so we can measure the lengths of AB, BC, and AC. Now one of
the side PQR is known accurately. Let us say the length of PQ is known. Hence, by using the
law of similar triangle
PIC2
Now, the value of K, BC and AC are known. Hence QR & PR can be found.
PIC 3
Let us consider triangle R and S are similar.
PIC 4
Right angle ∆ ACB and right angle triangle ∆ DFE are similar.
PIC 5
went out ten decimal places, and accuracy generally got worse as the numbers got larger. The
famous Eratosthenes experiment, that can be found
at https://www.famousscientists.org/eratosthenes/, relied on the fact known to Thales and others
that a beam of parallels cut by a transverse straight line determines 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.
The similarity of triangles gives rise to trigonometry.
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.
Your Discussion should be a minimum of 250 words in length and not more than 750 words.
_________________________________________________________________________________
Given two similar triangles
Let us say ∆ ABC ≈ ∆ PQR
PIC1
Now ABC is a small triangle so we can measure the lengths of AB, BC, and AC. Now one of
the side PQR is known accurately. Let us say the length of PQ is known. Hence, by using the
law of similar triangle
PIC2
Now, the value of K, BC and AC are known. Hence QR & PR can be found.
PIC 3
Let us consider triangle R and S are similar.
PIC 4
Right angle ∆ ACB and right angle triangle ∆ DFE are similar.
PIC 5
