2020 · Q73
A circle of diameter 8 cm is placed in such a manner that it touches two perpendicular lines. Then another smaller circle is placed in the gap such that it touches the lines and the circle. What is the diameter of the smaller circle?
Elementary MathematicsCircles
- A4(3 - √2) cm
- B4(3 - 2√2) cm
- C8(3 - √2) cm
- D8(3 - 2√2) cm
Explanation
The big circle's centre is at (4, 4), the small one's at (s, s) from the corner, and they touch externally: 4√2 = s√2 + (4 + s). Hence s = 4(√2 − 1)/(√2 + 1) = 4(3 − 2√2), and the diameter is 8(3 − 2√2) cm. Numerically the radius is about 0.686 cm, so the diameter is about 1.37 cm.