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3 years ago

The diagram shows the sector of a circle with the centre O and radius 6cm.

Mathematics

1 answer:

cricket20 [7]3 years ago

3 0

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 = $\frac{\theta}{360}(\pi r^{2})$

Here, θ = Central angle of the sector

r = radius of the sector

Area of sector OMN = $\frac{50}{360}(\pi )(6)^2$

= 15.708 square cm

Area of ΔOMN = 2(ΔOPN)

Area of ΔOPN = $\frac{1}{2}(OP)(PN)$

Area of ΔOMN = OP × PN

In ΔOPN,

sin(25°) = $\frac{PN}{ON}$

PN = ONsin(25°)

= 6sin(25°)

= 2.536 cm²

cos(25°) = $\frac{OP}{ON}$

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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