Problem 15 · 2012 Math Kangaroo
Hard
Geometry & Measurement
areaspatial-reasoning
One of the two sides of a rectangle has length 6 cm. In the rectangle circles are drawn next to each other in such a way that their centres form an equilateral triangle. What is the shortest distance between the two grey circles (in cm)?
Show answer
Answer: C — \(2\sqrt{3} - 2\)
Show hints
Hint 1 of 2
Three equal circles span the 6 cm side, so each has diameter 2 and radius 1.
Still stuck? Show hint 2 →
Hint 2 of 2
Equal touching circles have centres 2 apart; stack the equilateral rows to find the two grey centres.
Show solution
Approach: locate the grey centres, then subtract the two radii
- Three touching circles fill the 6 cm width, so each diameter is 2 and each radius is 1.
- Centres of touching circles are 2 apart and form equilateral triangles, so going down two rows the centres drop by 2 × √3 = 2√3 vertically.
- The two grey circles' centres are 2√3 apart; subtract the two radii (1 + 1 = 2) to get the gap.
- The shortest distance is 2√3 − 2 (C).
Mark:
· log in to save