ABCD is a rectangle in which AB=20cm and BC=10cm A semicircle is drawn with centre O and radius 10root2 - Maths - Areas Related to Circles

Question

ABCD is a rectangle in which AB=20cm and BC=10cm A semicircle is
drawn with centre O and radius 10root2 cm It passes through A and
,touching the ends Find the area of shaded region

Manbar Singh · Accepted Answer

In ∆OAB, OA = OB = radii of circle = 102 cmNow, AB = 20 cmNow, AB2 = 202 = 400and OA2 + OB2 = 1022 + 1022 = 200 +200 = 400Since, OA2 + OB2 =AB2⇒AOB is a right ∆ in which ∠AOB = 90°   Converse of Pythagoras TheoremTake any point X on the arc AB of the semicircle
Consider the sector OAXB,radius,r = 102 cm; central angle, θ = ∠AOB = 90°area of sector OAXB = θ360°×πr2 = 90°360°×π×200 = 50π cm2ar∆AOB = 12×OA×OB = 2002 = 100 cm2area of segment AXB = area of sector OAXB - ar∆AOB⇒area of segment AXB = 50π-100 cm2 area of shaded region = area of ABCD- area of segment AXB = 20×10 -50π-100  = 300-50π cm2

ABCD is a rectangle in which AB=20cm and BC=10cm. A semicircle is drawn with centre O and radius 10root2. cm. It passes through A and ,touching the ends. Find the area of shaded region .