The figure above shows 2 circles. The larger circle has center A, radius R cm, and is inscribed in a square. The smaller circle has center C, radius r cm, and is tangent to the larg circle at point B and to the square at points D and F. If points A, B, C, and E are collinear,which of the following is equal to R/r?
~$\frac{2}{\sqrt{2}+1}$~
~$\frac{2}{\sqrt{2}-1}$~
~$\frac{2}{2\sqrt{2}+1}$~
~$\frac{\sqrt{2}+1}{\sqrt{2}-1}$~
~$\frac{2\sqrt{2}+1}{2\sqrt{2}-1}$~
正方形的对角线一半长度=“大圆半径”+“小圆半径”+“小圆半径作边长对应的直角三角形斜边长”
R√2-R=r+r√2
AE是要分成三段来计算的
关键在于构造三角形:延长FC,作垂线AG与FC的延长线交于点G,连接AC。这样就形成了三角形ACG。可知▲ACG是一个等腰直角三角形,两腰长R-r,斜边长R+r。根据等腰直角三角形的边长比可知,AC: AG= (R+r) : (R-r) = √2 : 1
根号2*R=R+r+根号2*r
√2*(R - r) = R + r