Diagonal PR divides parallelogram PQRS into equal triangles PSR and PQR. Similarly, triangles OSR and OQR share base and equal heights, making their areas equal. Subtracting gives area(ΔPSO) = area(ΔPQO).
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Both triangles PCD and QCD share the common base CD and lie between parallel lines AB and CD, giving identical perpendicular heights. Hence their areas are equal, and the ratio area(ΔPCD) : area(ΔQCD) is 1:1.
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 ...
Ratio parts sum to 3 + 5 + 7 = 15, giving sides 60 m, 100 m, and 140 m with semi-perimeter s = 150 m. Area = √(150 x 90 x 50 x 10) = 1500√3 m² (approx 2598.08 ...
Third side is 32 – (8 + 11) = 13 cm. Semi-perimeter s = 32 / 2 = 16 cm. By Heron’s formula, area = √(16 x (16 – 8) x (16 – 11) x (16 – 13)) = √(16 ...