Question:
Three competing radio stations A, B and C are allowed to broadcast in a specific area.
The place where radio A is located is 47 km from radio B.
The place where radio B is located is 41 km from radio C.
The place where radio C is located is 22 km from radio A.
We represent their transmission range in the figure below. Determine the area of the transmission range of each radio station.
What I tried: I thought I'd try to find the corresponding points on a graph, but once I did that it didn't get clearer, I still couldn't find any relation between the areas of the circles and the triangle. Is there some kind of identity for triangles that have three circles that perfectly touch each other?
Im supposed to solve this with matrices.
The distance between each center can be written as sum of two radii and that will give me the system of 3 equations: $\begin{cases} R_A+R_B=47 \\ R_B+R_C=41 \\ R_A+R_C=22 \end{cases}$
The solutions are $(R_A,R_B,R_C)=(14,33,8)$ (I suggest using the substitution method)
What is left to do is to calculate the area of each circle:
$\begin{cases} A_A=\pi R_A^2=196\cdot3,14=615,44 \\ A_B=\pi R_B^2=1089\cdot3,14=3419,46 \\ A_C=\pi R_C^2=64\cdot3,14=200,96 \end{cases}$
All areas are expressed in square kilometers.