Problem Solving

There are two concentric circles. The chord to the larger one is also tangent to the smaller one. If chord length is 6, then find the difference in the radii of two circles.

Since the cord is tagent to circle 2 than the radius is perpendicular to the cord, and since the circles are concetric, then the triangle made up of half the cord, r1 and r2 is a right triangle (as seen in the attached figure)

so we can use pythagorean theorem:

r1^2+3^2=r2^2
r1^2-r2^2=9
(r1-r2)*(r1+r2)=9

And that's as far as I can go... anyone has a clue how to continue?

Well data is actually not sufficient to solve this question,but if the question comes in the Problem Solving section of GMAT i will approach like this:

r1^2 +3^2 = r2^2

this is in the form

4^2 + 3^2 = 5^2

thats the only way one can have integral value of R1 and R2

R2=5
R1 =4

R2-R1 = 5-4 = 1

Please Post the OA and the source of the question

Karan

gmat740 wrote:Well data is actually not sufficient to solve this question,but if the question comes in the Problem Solving section of GMAT i will approach like this:

r1^2 +3^2 = r2^2

this is in the form

4^2 + 3^2 = 5^2

thats the only way one can have integral value of R1 and R2

R2=5
R1 =4

R2-R1 = 5-4 = 1

Please Post the OA and the source of the question

Karan

OA is 1, integral value is not an imposed condition, let's wait and look for an explanation that doesn't mind the inadequacy of data here. Source of this question is vindictive geometry.

Could you please elabotrate it? how did you get 1?

hmboy17 wrote:Could you please elabotrate it? how did you get 1?

I didn't get 1; OA got 1!

The chord being tangential to the inner circle, tells us two things:

1) The line joining the center (0) and the chord's intersection with inner circle (T) is perpendicular with the chord (AB).

2) Triangle OAB is isoceles, because: OA = OB = outer circle's radius R

In an isoceles triangle, OT is also the bisector of AB; i.e. AT = TB = 3

Now we have a right angle triangle: OAT with OA = R and OT = inner circle radius r

Triangle OAT being a right angle triangle, allows to have a pythogorean triplet for its sides. One obvious Pyth triplet involving 3 is 3,4,5
So r = 4; R = 5
Hence |r-R| = 1