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steposvetlana [31]
2 years ago
9

Pls help 15. Find the exact area of the shaded region of the circle shown in the diagram.

Mathematics
1 answer:
konstantin123 [22]2 years ago
8 0

Answer:

  • The shaded region is 9.83 cm²

Step-by-step explanation:

<em>Refer to attached diagram with added details.</em>

<h2>Given </h2>

Circle O with:

  • OA = OB = OD - radius
  • OC = OD = 2 cm

<h2>To find</h2>

  • The area of segment ADB.

<h2>Solution</h2>

Since r = OC + CD, the radius is 4 cm.

Consider right triangles OAC or OBC:

  • They have one leg of 2 cm and hypotenuse of 4 cm, so the hypotenuse is twice the short leg.

Recall the property of 30°x60°x90° triangle:

  • a : b : c = 1 : √3 : 2, where a- short leg, b- long leg, c- hypotenuse.

It means OC: OA = 1 : 2, so angles AOC and BOC are both 60° as adjacent to short legs.

In order to find the shaded area we need to find the area of sector OADB and subtract the area of triangle OAB.

Area of <u>sector:</u>

  • A = π(θ/360)r², where θ- central angle,
  • A = π*((mAOC + mBOC)/360)*r²,
  • A = π*((60 + 60)/360))(4²) =  16.76 cm².

Area of<u> triangle AO</u>B:

  • A = (1/2)*OC*(AC + BC),  AC = BC = OC√3 according to the property of 30x60x90 triangle.
  • A = (1/2)(2*2√3)*2 = 4√3 = 6.93 cm²

The shaded area is:

  • A = 16.76 - 6.93 = 9.83 cm²

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