Way 1: Since 7² + 24² = 49 + 576 = 625 = 25², it is a right triangle, so Area = (1/2) x 7 x 24 = 84 cm². Way 2: Semi-perimeter s = 28 cm; Heron’s formula yields ...
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The square’s diagonal equals the circle’s diameter 2r. Area of the square is (1/2) x diagonal² = (1/2) x (2r)² = 2r². Dividing by circle area πr² gives the ratio 2 / π ≈ 0.637.
Minor sector is (60/360) x 3.14 x 15² = 117.75 cm². Equilateral triangle area is (√3/4) x 15² = 97.31 cm². Minor segment is 117.75 – 97.31 = 20.44 cm². Major segment is 706.5 – 20.44 = 686.06 cm².
Since CQ || PD, triangles DPQ and DPC share base PD and parallel lines, giving Area(ΔDPQ) = Area(ΔDPC). Thus, Area(ΔBPQ) = Area(ΔBPD) + Area(ΔDPQ) = Area(ΔBPD) + Area(ΔDPC) = Area(ΔBDC) = (1/2) x Area(ΔABC).
Let shorter diagonal be d, so longer diagonal is 2d. Area of rhombus is (1/2) x d1 x d2 = (1/2) x d x 2d = d². Equating d² = 128 gives shorter diagonal d = √128 = 8√2 cm ...