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diamong [38]
1 year ago
9

Find the area of the segment (unshaded area) of Circle G with radius 4in. Round to the nearest tenth.

Mathematics
2 answers:
frez [133]1 year ago
5 0

Answer:

4.6 in²  (nearest tenth)

Step-by-step explanation:

To find the <u>area of the unshaded region</u>, subtract the <u>area of ΔAGB</u> from the <u>area of sector AGB</u>.

The measure of an arc is equal to its corresponding central angle measure.  Therefore, the <u>central angle of sector</u> AGB is 90°.

As the two sides of ΔAGB adjacent the central angle are the radii of the circle they are therefore equal in length ⇒ ∠GAB = ∠GBA.  

Therefore, ΔAGB is an isosceles triangle.

<u>Area of triangle (using the Sine Rule):</u>

\sf A=\dfrac{1}{2}ab \sin C

(where a and b are the side lengths and C is the included angle)

Given:

  • a = b = radius = 4 in
  • C = 90°

\implies \sf Area\:of\:triangle=\dfrac{1}{2}(4)(4)\sin 90^{\circ}=8\:in^2

<u>Area of a sector of a circle</u>

\textsf{A}=\left(\dfrac{\theta}{360^{\circ}}\right) \pi r^2

\textsf{(where r is the radius and the angle }\theta \textsf{ is measured in degrees)}

Substituting the given angle and radius:

\implies \textsf{A}=\left(\dfrac{90^{\circ}}{360^{\circ}}\right) \pi (4)^2=4\pi\:\: \sf in^2

<u>Area of the unshaded region:</u>

\begin{aligned}\textsf{Area of unshaded region} & =\textsf{Area of sector} - \textsf{Area of triangle}\\& = 4 \pi - 8\\& = 4.566370614...\\ & = 4.6\:\sf in^2\:\:(nearest\:tenth)\end{aligned}

Romashka [77]1 year ago
4 0

Angle is 90°

The sector is 1/4 th of circle

Area of sector

  • πr²/4
  • π(4)²/4
  • 4πin²

Triangle is right angled

Area

  • 1/2(4)²
  • 16/2
  • 8in²

Area of unshaded region

  • 4π-8
  • 4(π-2)
  • 4(3.14-2)
  • 4(1.14)
  • 4.56in²
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