Trisection points divide the base into three equal segments, and both triangles share the same vertex altitude. Area equals (1/2) x base x height, so their areas are identical. Straight dissection along altitudes enables rearrangement.
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By Pythagoras theorem, leg semicircles sum to the hypotenuse semicircle: Semicircle(a) + Semicircle(b) = Semicircle(c). Subtracting the two unshaded circular segments from both sides shows that the lunes’ area Area(A) + Area(B) equals the triangle’s area Area(C).
Drawing a diagonal splits the trapezium into two triangles having bases a and b, each sharing perpendicular height h. Summing their areas gives (1/2)ah + (1/2)bh = (1/2)(a + b)h.
Circumference 2 x (22/7) x r = 66 gives radius r = 21/2 cm. The area of the quadrant is (1/4) x π x r² = (1/4) x (22/7) x (21/2)² = 693/8 = 86.625 cm².
Using area = (1/2) x base x height = 60 with base 10 cm gives height h = 12 cm. In the half-triangle right triangle with legs 5 cm and 12 cm, hypotenuse is √(5² + 12²) = 13 cm.