In triangle ABC, angle A = 90o, AB = 12 cm and BC = 20 cm. Three semi-circles are drawn with AB, AC and BC as diameters. Find the area of the shaded portion.



Asked by Topperlearning User | 27th Jul, 2017, 02:33: PM

Expert Answer:

 begin mathsize 12px style In space right space increment ABC comma
BC squared equals space AB squared space plus space AC squared space left parenthesis Pythagoras space theorem right parenthesis
rightwards double arrow space open parentheses 20 close parentheses squared space equals space open parentheses 12 close parentheses squared space plus space AC squared
rightwards double arrow space AC squared space equals space 400 space minus space 144 space equals space 256 space rightwards double arrow space AC space equals space 16 space cm
Area space of space shaded space region space equals space Area space of space semi minus circle space on space side space AB space plus space Area space of space semi minus circle space on space side space AC space
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space minus space Area space of space semi minus circle space on space side space BC space plus space Area space of space increment ABC
Area space of space semi minus circle space on space side space AB space equals space πr squared over 2 equals fraction numerator 22 over denominator 7 space cross times space 2 end fraction cross times space 6 space cross times space 6 space equals space 396 over 7 space cm squared
Area space of space semi minus circle space on space side space AC space equals space πr squared over 2 space equals space fraction numerator 22 over denominator 7 space cross times space 2 end fraction space cross times space 8 space cross times space 8 space equals space 704 over 7 space cm squared
Area space o f space semi minus circle space on space side space BC space equals space πr squared over 2 space equals space fraction numerator 22 over denominator 7 space cross times space 2 end fraction space cross times space 10 space cross times space 10 space equals space 1100 over 7 space cm squared
Area space of space increment ABC space equals space 1 half space cross times space AB space cross times space AC space equals space 1 half space cross times space 12 space cross times space 16 space equals space 96 space cm squared
Area space of space shaded space region space equals space 396 over 7 space plus space 704 over 7 space minus space 1100 over 7 space plus space 96 space cm squared space equals 0 plus 96 equals space 96 space cm squared
Hence comma space the space required space area space equals space 96 space cm squared end style

Answered by  | 27th Jul, 2017, 04:33: PM

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