the shape alongside is one quarter of a circle with radius of 14 cm. find the length of arc AB ,perimeter of the figure , area , area of triangle AOB , area of shaded segment.​

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

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.

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:

[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:

[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:

[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:

[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:

[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

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