Question

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

Answer 1

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