I'll leave my answer in the comments as usual.
Monday, May 08, 2023
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The problem gives you two circles that are touching, and that sit on the same plane, which is shown as a line. The smaller circle has a radius of 7; the larger circle has a radius of 21. How long is x, which is the distance between the two points of tangency on the line?
ReplyDeleteTo figure out the length of x, you need to construct a trapezoid. From each circle's center, draw a perpendicular segment down to the line, then draw a single line segment connecting the circle's two centers. You now have a trapezoid that's "on its side," so to speak.
The diagonal top of the trapezoid will have a length of 21; the left vertical side will have a height of 7; the right vertical side will have a height of 21; the bottom of the trapezoid will have a length of x, the thing you're solving for.
From the smaller circle's center, draw a line segment that extends to the right side of the trapezoid, parallel to the line at the bottom. By doing this, you have created a right triangle with a hypotenuse of 28, a right side of 14 (21 - 7 = 14), and a bottom of length x. To find x, then, all you need to do is apply the Pythagorean Theorem:
x^2 + 14^2 = 28^2, or
x^2 + 196 = 784, or
x^2 = 588. So,
x = √588, or 14√3. In decimals:
x ≈ 24.25.
QED.