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).
In Fig. 6.52, semicircles have been drawn on all the sides of a right-angled triangle as shown. Show that Area (A) + Area (B) = Area (C).
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Let the legs of the right-angled triangle C be a and b, and the hypotenuse be c.
Area of semicircle on leg a = (1/2) x π x (a/2)² = (1/8)πa².
Area of semicircle on leg b = (1/2) x π x (b/2)² = (1/8)πb².
Area of semicircle on hypotenuse c = (1/8)πc².
By Pythagoras theorem, a² + b² = c², so:
Area(Semicircle a) + Area(Semicircle b) = Area(Semicircle c).
The semicircle on hypotenuse c consists of triangle C plus two circular segments lying outside the legs.
The two lunes A and B are formed by subtracting these same two circular segments from the two smaller semicircles.
Therefore:
Area(A) + Area(B) = Area(Semicircle a) + Area(Semicircle b) – Segments
= Area(Semicircle c) – Segments = Area(C).
Hence proved.
For more NCERT Solutions of Class 9 Ganita Manjari Chapter 6 Measuring Space: Perimeter and Area Question Answer (2026-27)
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-6/