ok let's start the problem by first envisioning the two circles. First, we have circle a, with a radius of 10, and circle b with a radius of 7. If we connect AB and extend it into a line, we get the line AB. Drawing a line tangent to both circles creates a right triangle triangle, with the line intersecting line AB at C. Now, we have a big right trianlge with a right angle located at the place where the radius of circle A intersects the tangent line(remember the definition of tangent!) We can then form another right triangle by drawing the radius of circle B to intersect the tangent line. This gives us two similar right triangles(by AA similarity), with the ratio of the big triangle to small triangle being 10/7(the ratio of the radii). With this information in hand, we can solve for the length of the hypotenuse of the large triangle by using the similarity ratio:
if we name the length BC as x, we get 17+x : x = 10:7 , or (17+x)/x = (10/7). Cross multipling, we get 10x = 119 + 7x, simplifying to 3x = 119, or x = 119/3
