The rectangle is inscribed in a circle with Dia = 10 cm. The area of the segment that is formed by the shorter side of the rectangle and the circumference of a circle is 4 cm^{2}. Find the area of that rectangle.

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Guest Aug 30, 2019

edited by
Guest
Aug 30, 2019

edited by Guest Aug 30, 2019

edited by Guest Aug 31, 2019

edited by Guest Aug 30, 2019

edited by Guest Aug 31, 2019

#1**+1 **

The rectangle is inscribed in a circle with Dia = 10 cm. The area of the segment that is formed by the shorter side of the rectangle and the circumference of a circle is 4 cm2. Find the area of that rectangle.

**Hello Guest!**

\(\alpha\) = center angle

r = radius

s = chord

b = long side of the rectangle

A = segment area

\(A_R\) = rectangle area

\(\color{blue}A=\frac{r^2}{2}(\alpha -sin \alpha )\\ \alpha -sin \alpha=\frac{2\cdot A}{r^2}=\frac{8cm^2}{(5cm)^2}\\ \alpha -sin\ \alpha=0.32\\ \alpha -sin\ \alpha -0.32=0 \)

http://www.arndt-bruenner.de/mathe/scripts/gleichungssysteme2.htm

Solution found in 1st run after 3 iterations

\(\alpha = 1,27721309204\)

\(s=2r\cdot sin(\frac{\alpha}{2})\\ s=2\cdot 5cm\cdot sin(\frac{1.27721309204}{2})\\ \color{blue}s=5.961cm\)

\(cos(\frac{\alpha}{2})=\frac{\frac{b}{2}}{r}\\ b=2r\cdot cos(\frac{\alpha}{2})=10cm\cdot cos(\frac{1,27721309204}{2})\)

\(b=8.029cm\)

\(A_R=s\cdot b=5.961cm\cdot 8.029cm\)

\(A_R=47.862\ cm^2\)

!

asinus

Guest Aug 31, 2019

#3**+2 **

Let θ be the central angle whose endpoints are on the shorter side of the rectangle

The area of the sector formed by this angle = ( r^2/2 ) ( θ)

And the area of the isosceles triangle formed by the two radii and the shorter side of the rectangle =

(r^2/2) sin ( θ)

So....the area between the shorter side of the rectangle and the circumference is given by

4 = (r/2/2) ( θ - sin θ)

4 = (5^2/2) ( θ -sin θ)

4 = (25/2) ( θ - sin θ)

θ - sin θ = 8/25

Using a little technology to solve this for θ ≈ 1.27721 rads ≈ 73.18°

From the Law of Cosines we can find the shorter side of the rectangle as

√ [ 5^2 + 5^2 - 2(5)(5)cos(73.18°) ] = √[50 -50cos(73.18°)] ≈ 5.96 cm

And using the Pythagorean Theorem, we can find the longer side of the rectangle as

√[10^2 - 5.96^2 ] ≈ 8.03 cm

So.....the area of the rectangle will be the product of these two sides ≈ 5.96 * 8.03 ≈ 47.86 cm^2

Here's a pic :

CPhill Aug 31, 2019