Problem 19 · 1994 AJHSME
Hard
Geometry & Measurement
tangentdimensions
Show answer
Answer: E — 64.
Show hints
Hint 1 of 2
A semicircle drawn on a side of length 4 has that side as its diameter, so its radius is 2 — and the farthest it pokes out from the side is exactly that radius, 2.
Still stuck? Show hint 2 →
Hint 2 of 2
The outer square is tangent to the bulges, meaning it just kisses the top of each semicircle. So measuring straight across, the outer side = inner side + a bulge on each end. Then square it for area.
Show solution
Approach: grow the inner square by the semicircle radii
- Diameter = the side = 4, so each semicircle's radius is 2. Its outermost point sticks out 2 past the inner square's edge.
- Cross the figure left-to-right: outer edge, bulge (2), inner square (4), bulge (2), outer edge. So square ABCD has side 4 + 2 + 2 = 8.
- Area = 8² = 64.
- Sanity check that beats the trap answers: the outer square is clearly bigger than the inner one's area (16) but not absurdly so — 64 = four inner squares, which looks right in the picture. The takeaway: when a shape 'just touches' a circle (tangency), the gap it leaves equals the radius — convert tangency into a length you can add.
Mark:
· log in to save