This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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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^. Use = (22)/(7). Step 1: Find the radius of the circle. The diameter is 42 cm, so the radius r is half of the diameter. r = Diameter2 = 42 cm2 = 21 cm Step 2: Find the minor angle ROT. The sum of angles around a point is 360^. Minor ROT = 360^ - Reflex ROT Minor ROT = 360^ - 240^ = 120^. Step 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^ and OTP = 90^. The sum of angles in a quadrilateral is 360^. In quadrilateral ORPT: RPT + ORP + OTP + Minor ROT = 360^ RPT + 90^ + 90^ + 120^ = 360^ RPT + 300^ = 360^ RPT = 360^ - 300^ = 60^. Step 4: Calculate the length of the tangents PR and PT. Consider the right-angled triangle ORP. We know ORP = 90^, OR = r = 21 cm, and ROP = Minor ROT2 = (120^)/(2) = 60^. Using trigonometry: ( ROP) = (PR)/(OR) (60^) = (PR)/(21) PR = 21 × (60^) = 21 × sqrt(3) cm. PR ≈ 21 × 1.732 = 36.372 cm. Since tangents from an external point to a circle are equal, PT = PR = 21sqrt(3) cm. Step 5: Calculate the area of the quadrilateral ORPT. The quadrilateral ORPT can be divided into two congruent right-angled triangles, ORP and OTP. Area of ORP = (1)/(2) × base × height = (1)/(2) × OR × PR Area of ORP = (1)/(2) × 21 cm × 21sqrt(3) cm = 441sqrt(3)2 cm^2. Area of quadrilateral ORPT = 2 × Area of ORP = 2 × 441sqrt(3)2 cm^2 = 441sqrt(3) cm^2. 441sqrt(3) ≈ 441 × 1.732 = 763.812 cm^2. Step 6: Calculate the area of the minor sector ROT. Area of sector = ()/(360^) × r^2 Area of minor sector ROT = (120^)/(360^) × (22)/(7) × (21 cm)^2 Area of minor sector ROT = (1)/(3) × (22)/(7) × 441 cm^2 Area of minor sector ROT = (1)/(3) × 22 × 63 cm^2 Area of minor sector ROT = 22 × 21 cm^2 = 462 cm^2. Step 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 = 441sqrt(3) cm^2 - 462 cm^2 Area of shaded region ≈ 763.812 cm^2 - 462 cm^2 = 301.812 cm^2. Rounding to two decimal places: Area of shaded region ≈ 301.81 cm^2. What's next?