SMC2012_web_solutions

22 a semicircle of radius r is drawn with centre v

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22. A semicircle of radius r is drawn with centre V and diameter UW. The line UW is then extended to the point X, such that the UW and WX are of equal length. An arc of the circle with centre X and radius 4 r is then drawn so that the line XY is a tangent to the semicircle at Z, as shown. What, in terms of r, is the area of triangle YVW? A 9 4 2 r B 3 2 2 r C 2 r D 3 4 2 r E 2 2 r Solution: B Since WX UW = , we have . 2 r UW WX = = As V is the centre of the circle with UW as diameter, . r VZ VW = = Hence r WX VW VX 3 = + = . The triangles YVX and VVW have the same height and VX VW 3 1 = . Therefore ) ( area ) ( area 3 1 YVX YVW = . As VZ is a radius of the semicircle, and XY is a tangent to the semicircle at Z , the lines VZ and XY are perpendicular. Therefore . 2 ) 4 . ( . ) ( area 2 2 1 2 1 r r r XY VZ YVX = = = Hence, . ) 2 ( ) ( area ) ( area 2 3 2 2 3 1 3 1 r r YVX YVW = = = U V W X Y Z U V W X
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