Answer:
1) The length of the arc AB is approximately 21.99 cm
2) The perimeter of the figure is approximately 49.99 cm
3) The area of the figure is approximately 153.938 cm²
4) The area of the right triangle ΔAOB is 98 cm²
5) The area of the shaded segment is approximately 55.938 cm²
Step-by-step explanation:
The given parameters are;
The radius of the circle from which we have the quarter circle, r = 14 cm
Therefore, we have;
The length of segment OA = The length of segment OB = r = 14 cm
1) The length of the arc AB = 1/4 × The circumference of the circle with radius 14 cm and center O
The length of the arc AB = 1/4 × 2 × π × r = 1/4 × 2 × π × 14 c m= 7·π cm ≈ 21.99 cm which is approximately 22 cm, to the nearest whole number
2) The perimeter of the figure = The length of the arc AB + The length of segment OA + The length of segment OB
∴ The perimeter of the figure = 7·π cm + 14 cm + 14 cm ≈ 49.99 cm which is approximately 50 cm, to the nearest whole number
3) The area of the figure (sector AOB) = The area of the quarter of a circle = π × r²/4 = π × (14 cm)²/4 = 49·π cm² ≈ 153.938 cm²
4) Whereby we have arc AB and segment OA and OB form a quarter (1/4) of a circle, we have;
∠AOB = 360°/4 = 90°
Therefore, ΔAOB is a right triangle
The base length of the right triangle ΔAOB (is taken as being) = Segment OA = 14 cm
The height of the right triangle ΔAOB (is taken as being) = Segment OA = 14 cm
The area of the right triangle ΔAOB = 1/2 × The base length × The height
∴ The area of the right triangle ΔAOB = 1/2 × 14 cm × 14 cm = 98 cm²
The area of the right triangle ΔAOB = 98 cm²
5) The area of the shaded segment = The area of the quarter of a circle (Sector OAB) - The area of the triangle ΔAOB
The area of the quarter of a circle = 49·π cm² ≈ 153.938 cm²
∴ The area of the shaded segment = 49·π cm² - 98 cm² ≈ 55.938 cm²
The area of the shaded segment ≈ 55.938 cm²