Question about calculating the length of an arc, circle circumscribed around a scalene triangle.

Question

1. Question

My question concerns the area of geometry and trigonometry, specifically involving an arc of a semicircle with a circle circumscribed around a scalene triangle. I am trying to calculate the lenght of the arc (in yellow line), which, according to the answer section of the book for this exercise, is approximately 937.72 meters, but without any resolution as a complement.

Representation of the exercise.

Initial values include:
a= 473.79095176625214 meters.
b=436.57425985962715 meters.
c=687.004949600736 meters.
h=298.2205425946874 meters
C-D= 368.1607990404481 meters.
D-A= 318.8441505602878 meters.

(!) Note that point B is not in the middle of the curvature segment, although it might appear that it is.

2. Discussion

My method of solving for this exercise, which proved to be apparently erroneous, was formulated as follows:

I first sought to calculate the area of the scalene triangle (S) by Heron's formula [1.], and then applied it in another form to find the radius of the circle [2.] With the value of the radius, it was possible to calculate the value of the apothem (t) [3.]. As a result of these processes, I had the value of the hypotenuse (represented by the radius) and a common side of the two triangles formed (represented by the apothem), and so I calculated the arc cosine to find the angle associated with the central point of the circle (represented by E), and then multiplying by two, since they are two triangles with the same values of hypotenuse and adjacent vector [4.]. In this way I just used the formula for calculating the lenght of the arc with this resulting angle [5].

The result of this calculation was formulated as approximately 990 meters, a value close to the desired one, but incompatible with the answer in the answer section of the book.

Is there a method of calculating the lenght of this arc, with this data, that comes close to the answer of 937.72 meters? Or is the final answer in the book wrong? Thank you in advance for any help, and apologies for any errors during my calculation method, open to understanding them. :)

Representation of the method

$$S\:=\sqrt{p.\left(p\:−\:a\right).\left(p\:−\:b\right).\left(p\:−\:c\right)}$$ [1.]

$$R=\frac{abc}{4S}$$ [2.]

$$t=R-h$$ [3.]

$$cosE=\frac{adj}{hyp}\:\:;\:\:2\left(cosE=\frac{t}{S}\right)$$ [4.]

$$L=2\pi r\cdot \left(\frac{\theta }{^{360^{\circ }}}\right)$$ [5.]

Answer 1

There is a much more efficient method to get the length of this arc.

Sines law says that

$$\frac{a}{\sin \hat A}=\frac{b}{\sin \hat B}=\frac{c}{\sin \hat C}=2r$$

where $r$ is precisely the radius of the circumscribed circle.

So the length of the arc is $\pi r$.

To get the value of $r$ you should use the cosines law. For example:

$$\cos\hat A=\frac{b^2+c^2-a^2}{2bc}$$ $$\sin\hat A=\sqrt{1-\cos^2\hat A}$$ $$\text{Length of the arc}=\frac{a\pi}{2\sin\hat A}$$

To get the precisest result with a calculator, choose $\hat A$ to be the angle with the greatest sine, that is the angle that is nearest to be right. (in general, avoid small denominators to make numerical computations).