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 4ris 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 942rB 322rC 2rD 342rE 22rSolution: B Since WXUW=, we have .2rUWWX==As V is the centre of the circle with UW as diameter, .rVZVW==Hence rWXVWVX3=+=. The triangles YVX and VVW have the same height and VXVW31=. Therefore )(area)(area31YVXYVW∆=∆. 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.(.)(area22121rrrXYVZYVX===∆Hence, .)2()(area)(area23223131rrYVXYVW==∆=∆U V W X Y Z U V W X
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