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Geometry Level 3

Perpendicular cords A C AC and D E DE intersect at B B so that the lengths of A B , B C AB,BC and B D BD are 3, 4 and 2 respectively. If the radius of the circle can be written as m n \dfrac{\sqrt m} n , where m m and n n are positive integers with m m square-free. Find m + n m+n .


The answer is 67.

Marta Reece by Marta Reece , 69, USA

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2 solutions

Marta Reece
May 13, 2016

Triangles ABE and DBC are similar, so the distance BE is 6. Diameter KL parallel to the line AC bisects cord DE giving FE = FD = 4. Similarly vertical diameter bisects AC and this results in the width of the rectangle BFGH being 1/2. The triangles KFE and DFL are also similar, therefore

K F 4 = 4 F L \frac {KF}{4} = \frac {4}{FL}

R − 1 2 4 = 4 R + 1 2 \frac {R-\frac{1}{2}}{4} = \frac {4}{R+\frac{1}{2}}

Solving for R we get

R = 65 2 R=\frac{\sqrt{65}}{2}

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The problem can be solved using analytical geometry as well.

Set your coordinate system with a center at point B, x-axis going to the right, past point C. This gives you the points A = (-3, 0), C = (4, 0), and D = (0, -2). Center of the circle is equidistant from all of these points. To be equidistant from both A and C it has to have an x-coordinate equal to 1/2. To be equidistant form D and C, it has to be on a perpendicular bisector of CD. Line through CD has a slope 1/2. The midpoint between C and D is (2, -1). Line through (2, -1) with a slope -2 is given by

y − ( − 1 ) = − 2 ( x − 2 ) y-(-1)=-2(x-2)

Substituting into it x = 1/2 we get the y-coordinate of the center of the circle as y = 2. Distance between the center, (1/2, 2) and (0, -2) is

( 1 2 ) 2 + 4 2 = 65 2 \sqrt{(\frac{1}{2})^2+4^2}=\frac{\sqrt{65}}{2}

Marta Reece - 5 years, 1 month ago
Roger Erisman
May 14, 2016

Recognizing that AB * BC = DB * BE or 3 * 4 =2 * BE gives BE =6

Perpendicular bisectors of the two chords define center O.

Perpendicular from O to AC connects to AC 3.5 from A.

O is 4 - 2 = 2 away from AC.

So we have a right triangle with sides 3.5 and 2 and hypotenuse R.

R^2 = 2^2 + (7 / 2)^2 = 4 + 49 / 4 = 16 / 4 + 49 / 4 = 65 / 4

R = 6 5 2 \dfrac{\sqrt 65} 2 so 65 + 2 = 67

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