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प्रश्न
In figure, ABCD is a trapezium with AB || DC, AB = 18 cm, DC = 32 cm and distance between AB and DC = 14 cm. If arcs of equal radii 7 cm with centres A, B, C and D have been drawn, then find the area of the shaded region of the figure.
उत्तर
AB = 18 cm,
DC = 32 cm
Distance between AB and DC = Height = 14 cm
Now, Area of the trapezium
= `1/2 xx "Sum of parallel sides" xx "Height"`
= `1/2 xx 18 + 32 xx 14`
= 350 cm2
As AB ∥ DC,
∴ ∠A + ∠D = 180°
And ∠B + ∠C = 180°
Also, radius of each arc = 7 cm
Therefore,
Area of the sector with central angle A = `1/2 xx (∠"A")/180^circ xx π xx "r"^2`
Area of the sector with central angle D = `1/2 xx (∠"D")/180^circ xx π xx "r"^2`
Area of the sector with central angle B = `1/2 xx (∠"B")/180^circ xx π xx "r"^2`
Area of the sector with central angle C = `1/2 xx (∠"C")/180^circ xx π xx "r"^2`
Total area of the sectors,
= `(∠"A")/360^circ xx pi xx "r"^2 + (∠"D")/360^circ xx pi xx "r"^2 + (∠"B")/360^circ xx pi xx "r"^2 + (∠"C")/360^circ xx pi xx "r"^2`
= `((∠"A" + ∠"D")/360^circ xx pi xx "r"^2) + ((∠"B" + ∠"C")/360^circ xx pi xx "r"^2)`
= `(180^circ/360^circ xx 22/7 xx 49) + (180^circ/360^circ xx 22/7 xx 49)`
= 77 + 77
= 154
∴ Area of shaded region
= Area of trapezium – (Total area of sectors)
= 350 – 154
= 196 cm2
Hence, the required area of shaded region is 196 cm2.
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