3 circles in an iso. trap.

Geometry Level 3

Find A B C D \frac{AB}{CD}


The answer is 7.

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1 solution

Let the lower left circle touches the line A B \overline {AB} at H H , mid point of A B \overline {AB} be G G , mid point of C D \overline {CD} be F F and A C \overline {AC} and B D \overline {BD} meet at E E . Let A H = x |\overline {AH}|=x and E F = y |\overline {EF}|=y . Then r x = 1 2 \dfrac{r}{x}=\dfrac{1}{2} , or x = 2 r x=2r . Again y + 4 r 7 r 2 = 4 3 \dfrac{y+4r}{\dfrac{7r}{2}}=\dfrac{4}{3} , or y = 2 r 3 y=\dfrac{2r}{3} . Also, y C D = y + 4 r 7 r \dfrac{y}{|\overline {CD}|}=\dfrac{y+4r}{7r} . Therefore C D = r |\overline {CD}|=r , and the required ratio is 7 r r = 7 \dfrac{7r}{r}=\boxed 7

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