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Using pi = 22/7, find the area of the shaded region.

Mathematics

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Answer

301.81 cm2301.81 \text{ cm}^2

Step-by-step solution

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Here's the solution for question 4:

Given Information:

  • O is the center of the circle.
  • PR and PT are tangents to the circle at R and T respectively.
  • Diameter of the circle = 42 cm.
  • Reflex ∠ROT=240∘\angle ROT = 240^\circ.
  • Use π=227\pi = \frac{22}{7}.
  1. Find the radius of the circle.

    The diameter is 42 cm, so the radius rr is half of the diameter.

    r=Diameter2=42 cm2=21 cmr = \frac{\text{Diameter}}{2} = \frac{42 \text{ cm}}{2} = 21 \text{ cm}
  2. Find the minor angle ∠ROT\angle ROT.

    The sum of angles around a point is 360∘360^\circ.
    Minor ∠ROT=360∘−Reflex ∠ROT\angle ROT = 360^\circ - \text{Reflex } \angle ROT
    Minor ∠ROT=360∘−240∘=120∘\angle ROT = 360^\circ - 240^\circ = 120^\circ.

  3. Find the angles in quadrilateral ORPT.

    Since PR and PT are tangents to the circle at R and T, the radii OR and OT are perpendicular to the tangents at the points of tangency.
    Therefore, ∠ORP=90∘\angle ORP = 90^\circ and ∠OTP=90∘\angle OTP = 90^\circ.
    The sum of angles in a quadrilateral is 360∘360^\circ.
    In quadrilateral ORPT:
    ∠RPT+∠ORP+∠OTP+Minor ∠ROT=360∘\angle RPT + \angle ORP + \angle OTP + \text{Minor } \angle ROT = 360^\circ
    ∠RPT+90∘+90∘+120∘=360∘\angle RPT + 90^\circ + 90^\circ + 120^\circ = 360^\circ
    ∠RPT+300∘=360∘\angle RPT + 300^\circ = 360^\circ
    ∠RPT=360∘−300∘=60∘\angle RPT = 360^\circ - 300^\circ = 60^\circ.

  4. Calculate the length of the tangents PR and PT.

    Consider the right-angled triangle △ORP\triangle ORP.
    We know ∠ORP=90∘\angle ORP = 90^\circ, OR=r=21 cmOR = r = 21 \text{ cm}, and ∠ROP=Minor ∠ROT2=120∘2=60∘\angle ROP = \frac{\text{Minor } \angle ROT}{2} = \frac{120^\circ}{2} = 60^\circ.
    Using trigonometry:
    tan⁡(∠ROP)=PROR\tan(\angle ROP) = \frac{PR}{OR}
    tan⁡(60∘)=PR21\tan(60^\circ) = \frac{PR}{21}
    PR=21×tan⁡(60∘)=21×3 cmPR = 21 \times \tan(60^\circ) = 21 \times \sqrt{3} \text{ cm}.
    PR≈21×1.732=36.372 cmPR \approx 21 \times 1.732 = 36.372 \text{ cm}.
    Since tangents from an external point to a circle are equal, PT=PR=213 cmPT = PR = 21\sqrt{3} \text{ cm}.

  5. Calculate the area of the quadrilateral ORPT.

    The quadrilateral ORPT can be divided into two congruent right-angled triangles, △ORP\triangle ORP and △OTP\triangle OTP.
    Area of △ORP=12×base×height=12×OR×PR\triangle ORP = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times OR \times PR
    Area of △ORP=12×21 cm×213 cm=44132 cm2\triangle ORP = \frac{1}{2} \times 21 \text{ cm} \times 21\sqrt{3} \text{ cm} = \frac{441\sqrt{3}}{2} \text{ cm}^2.
    Area of quadrilateral ORPT = 2×Area of △ORP=2×44132 cm2=4413 cm22 \times \text{Area of } \triangle ORP = 2 \times \frac{441\sqrt{3}}{2} \text{ cm}^2 = 441\sqrt{3} \text{ cm}^2.
    4413≈441×1.732=763.812 cm2441\sqrt{3} \approx 441 \times 1.732 = 763.812 \text{ cm}^2.

  6. Calculate the area of the minor sector ROT.

    Area of sector = θ360∘×πr2\frac{\theta}{360^\circ} \times \pi r^2
    Area of minor sector ROT = 120∘360∘×227×(21 cm)2\frac{120^\circ}{360^\circ} \times \frac{22}{7} \times (21 \text{ cm})^2
    Area of minor sector ROT = 13×227×441 cm2\frac{1}{3} \times \frac{22}{7} \times 441 \text{ cm}^2
    Area of minor sector ROT = 13×22×63 cm2\frac{1}{3} \times 22 \times 63 \text{ cm}^2
    Area of minor sector ROT = 22×21 cm2=462 cm222 \times 21 \text{ cm}^2 = 462 \text{ cm}^2.

  7. Calculate the area of the shaded region.

    The shaded region is the area of the quadrilateral ORPT minus the area of the minor sector ROT.
    Area of shaded region = Area of quadrilateral ORPT - Area of minor sector ROT
    Area of shaded region = 4413 cm2−462 cm2441\sqrt{3} \text{ cm}^2 - 462 \text{ cm}^2
    Area of shaded region ≈763.812 cm2−462 cm2=301.812 cm2\approx 763.812 \text{ cm}^2 - 462 \text{ cm}^2 = 301.812 \text{ cm}^2.

    Rounding to two decimal places:
    Area of shaded region ≈301.81 cm2\approx \boxed{301.81 \text{ cm}^2}.

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