More generally:
Consider two circles with radii then the value of line externally tangent to both circles is given by
2 circles – one of radius 5, the other of radius 8 – intersect at exactly one point and the center of each circle lies outside the other circle. A line is externally tangent to both circles. Find the distance between the two points of tangency. Draw a figure and label it. Justify your answer completely with postulates and theorems. A two-column proof is suggested.
THANKS SO MUCH FOR HELPING ME!!
Hello,
I've attached a diagram of the situation. (The light grey circles are necessary to construct the tangent points)
To calculate the distance between the tangent points you are dealing with a right triangle, which I've coloured grey.
The length of the hypotenuse is (R + r)
One leg (coloured violet) has the length (R-r)
Now use Pythagorean theorem:
Expand the brackets:
. Subtract on both sides r² and R²: