Intersecting Circles

You may need a diagram of the two circles intersecting for this solution. Let's draw two circles with radii $R_1$ and $R_2$ which intersect at $A$ and $B$ . If you connect the centres and join point $A$ to the centres of the circle, you get a triangle. The angle that is connected to the centre of the circle with radius $R_1$ will be called $\theta_1$ and angle in the circle with radius $R_2$ will be called $\theta_2$ .

Just to note, one of the circles needs to have a radius of 50 at least, otherwise it isn't possible.

Now we need to use trigonometry.

$sin \theta_1 = \frac {50}{R_1}$

$R_1 = 50 \cosec \theta_1$

$sin \theta_2 = \frac {50}{R_2}$

$R_2 = 50 \cosec \theta_2$

$R_1 + R_2 = 50 (\cosec \theta_1 + \cosec \theta_2$

$R_1 + R_2 = 50 (1 + 1)$

$R_1 + R_2 = 100$

100 is the answer. Note: the highest value of any trigonometric function is 1, hence $\cosec \theta$ values were equalling 1.