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The diagram shows the sector of a circle with the centre O and radius 6cm.
MN is a chord of the angle.
angle MON is 50 degrees

calculate the area of the shaded segment
give your answer to 3 significant figures


The Diagram Shows The Sector Of A Circle With The Centre O And Radius 6cm MN Is A Chord Of The Angle Angle MON Is 50 Degrees Calculate The Area Of The Shaded Se class=

Sagot :

Answer:

Area of the shaded region = 1.92 cm²

Step-by-step explanation:

From the picture attached,

Area of the shaded region = Area of the sector OMN - Area of the triangle OMN

Area of sector OMN = [tex]\frac{\theta}{360}(\pi r^{2})[/tex]

Here, θ = Central angle of the sector

r = radius of the sector

Area of sector OMN = [tex]\frac{50}{360}(\pi )(6)^2[/tex]

                                  = 15.708 square cm

Area of ΔOMN = 2(ΔOPN)

Area of ΔOPN = [tex]\frac{1}{2}(OP)(PN)[/tex]

Area of ΔOMN = OP × PN

In ΔOPN,

sin(25°) = [tex]\frac{PN}{ON}[/tex]

PN = ONsin(25°)

     = 6sin(25°)

     = 2.536 cm²

cos(25°) = [tex]\frac{OP}{ON}[/tex]

OP = ONcos(25°)

OP = 6cos(25°)

OP = 5.438 cm

Area of ΔOMN = 2.536 × 5.438

                         = 13.791 cm²

Area of the shaded region = 15.708 - 13.791 = 1.917 cm²

                                             ≈ 1.92 cm²

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