The area of the shaded segment is approximately 55.938 square centimeters.
The length of the arc AB is approximately 21.991 centimeters.
The perimeter of the figure is approximately 49.991 centimeters.
The area of triangle AOB is 98 square centimeters.
The area of the circle segment is approximately 153.938 square centimeters.
Procedure - Area of the shaded segment
The area of the shaded segment ([tex]A[/tex]), in square centimeters, is obtained by subtracting the area of the triangle from the area of the circle quarter, that is:
Area of the shaded segment
[tex]A = \frac{\pi\cdot r^{2}}{4}-\frac{1}{2}\cdot \left(\frac{\sqrt{2}}{2}\cdot r \right)\cdot \left(\sqrt{2}\cdot r\right)[/tex]
[tex]A = \frac{\pi\cdot r^{2}}{4}-\frac{1}{2}\cdot r^{2}[/tex]
[tex]A = \left(\frac{\pi}{4}-\frac{1}{2} \right)\cdot r^{2}[/tex] (1)
If we know that [tex]r = 14\,cm[/tex], then the area of the shaded segment is:
[tex]A = \left(\frac{\pi}{4}-\frac{1}{2} \right)\cdot (14\,cm)^{2}[/tex]
[tex]A \approx 55.938\,cm^{2}[/tex]
Where [tex]r[/tex] is the radius of the circle segment, in centimeters.
The area of the shaded segment is approximately 55.938 square centimeters. [tex]\blacksquare[/tex]
The length of the arc AB ([tex]s[/tex]) is determined by definition of circular arc:
Length of the arc AB
[tex]s = \frac{\pi}{2} \cdot r[/tex] (2)
If we know that [tex]r = 14\,cm[/tex], then the length of the arc AB is:
[tex]s = \frac{\pi}{2} \cdot (14\,cm)[/tex]
[tex]s \approx 21.991\,cm[/tex]
The length of the arc AB is approximately 21.991 centimeters. [tex]\blacksquare[/tex]
The perimeter of the figure ([tex]p[/tex]), in centimeters, is the sum of the arc AB and the two legs of the right triangle, that is:
Perimeter of the figure
[tex]p = \frac{\pi}{2}\cdot r + 2\cdot r[/tex]
[tex]p = \left(\frac{\pi}{2} + 2 \right)\cdot r[/tex] (3)
If we know that [tex]r = 14\,cm[/tex], then the perimeter of the figure is:
[tex]p = \left(\frac{\pi}{2} + 2 \right)\cdot (14\,cm)[/tex]
[tex]p \approx 49.991\,cm[/tex]
The perimeter of the figure is approximately 49.991 centimeters. [tex]\blacksquare[/tex]
The area of the triangle ([tex]A[/tex]) was used in the first part of this question and we proceed to extract it below:
Area of triangle AOB
[tex]A = \frac{1}{2}\cdot r^{2}[/tex] (4)
If we know that [tex]r = 14\,cm[/tex], then the area of the triangle AOB is:
[tex]A = \frac{1}{2}\cdot (14\,cm)^{2}[/tex]
[tex]A = 98\,cm^{2}[/tex]
The area of triangle AOB is 98 square centimeters. [tex]\blacksquare[/tex]
The area of the circle segment ([tex]A[/tex]) was used in the first part of this question and we proceed to extract it below:
Area of circle segment
[tex]A = \frac{\pi\cdot r^{2}}{4}[/tex] (5)
If we know that [tex]r = 14\,cm[/tex], then the area of the circle segment is:
[tex]A = \frac{\pi\cdot (14\,cm)^{2}}{4}[/tex]
[tex]A \approx 153.938\,cm^{2}[/tex]
The area of the circle segment is approximately 153.938 square centimeters. [tex]\blacksquare[/tex]
To learn more on circles, we kindly invite to check this verified question: https://brainly.com/question/11833983
