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Assoli18 [71]
2 years ago
15

Find the area of the shaded region. geometry please help if your good at it. will mark brainlist

Mathematics
1 answer:
AnnZ [28]2 years ago
8 0

Area of shaded region = <em>area of circle</em> - <em>area of segment</em>

(where "segment" refers to the unshaded region)

<em>Area of circle</em> = <em>π</em> (11.1 m)² ≈ 387.08 m²

The area of the segment is equal to the area of the sector that contains it, less the area of an isosceles triangle:

<em>Area of segment</em> = <em>area of sector</em> - <em>area of triangle</em>

<em />

130° is 13/36 of a full revolution of 360°.  This is to say, the area of the sector with the central angle of 130° has a total area equal to 13/36 of the total area of the circle, so

<em>Area of sector</em> = 13/36 <em>π</em> (11.1 m)² ≈ 139.78 m²

Use the law of cosines to find the length of the chord (the unknown side of the triangle, call it <em>x</em>) :

<em>x</em> ² = (11.1 m)² + (11.1 m)² - 2 (11.1 m)² cos(130°)

<em>x</em> ² ≈ 404.82 m²

<em>x</em> = 20.12 m

Call this length the base of the triangle. Use a trigonometric relation to determine the corresponding altitude/height, call it <em>y</em>. With a vertex angle of 130°, the two congruent base angles of the triangle each measure (180° - 130°)/2 = 25°, so

sin(25°) = <em>y</em> / (11.1 m)

<em>y</em> = (11.1 m) sin(25°)

<em>y</em> ≈ 4.69 m

Then

<em>Area of triangle</em> = <em>xy</em>/2 ≈ 1/2 (20.12 m) (4.69 m) ≈ 47.19 m²

so that

<em>Area of segment</em> ≈ 139.78 m² - 47.19 m² ≈ 92.59 m²

Finally,

Area of shaded region ≈ 387.08 m² - 92.59 m² ≈ 294.49 m²

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