Geometry_2－快遞百科

標題:

Geometry

發問:

In the figure below: 圖片參考：http://i1191.photobucket.com/albums/z467/robert1973/Crazygeom1-1_zpsda63e7e8.jpg AD is a diameter of the circle with length 20. XY is a straight line with BC being a part of it and BC is a chord of the circle with length 12, Also AX and DY are both perp. to XY. Find AX + DY

最佳解答:

Let M be the mid-pt of BC, O be the centre of the circle, then OM ⊥ BC (line joining centre and mid-pt. of chord ⊥ chord), this gives AX, OM and DY are parallel to each other Note that OB=OC=10 (radii) since M is the mid-point of BC, BM=12/2=6 OM2=OB2-BM2 (Pyth. thm.) OM2=(10)2- (6)2 = 64 OM = 8 by intercepting theorem, since OA=OD, AX//OM//DY, we have M is the mid-pt of XY, ie, 2MY=2XM= XY consider slope of AD = slope of OA slope of AD = (DY-AX)/(XY) = (DY-AX)/(2XM) --(1) slope of OD = (OM-AX)/(XM) --(2) combine (1) and (2) gives DY-AX = 2(OM-AX) rearranging gives DY+AX = 2OM = 2*8 = 16 --## ---------------------------------------------------------------------------------------------------------- actually, i did it mentally (since it used to be a MC question in 2010 HKCEE Mathematics(II) ): rotate the given figure(fig 1) by 180° about the centre of the circle to give fig2, superimpose the two figures, then it will be obvious to conclude that DY+AX = 2OM