Problem 10 · 2024 Math Kangaroo
Medium
Geometry & Measurement
area-decomposition
There are black and dashed paths in a park. Both paths divide the area of the park exactly in half. Which of the following statements about the areas of the sections A, B and C (shown in the diagram) is definitely correct?
Show answer
Answer: B — B = A + C
Show hints
Hint 1 of 2
Both paths cut the park into two equal halves, so the area on one side of a path equals the area on the other side.
Still stuck? Show hint 2 →
Hint 2 of 2
Compare the two halves: the regions where the paths disagree are exactly A, B and C, and that balance forces a relation between them.
Show solution
Approach: balance the two equal halves where the paths disagree
- The solid path puts area \(\tfrac{1}{2}\) of the park on each side; the dashed path does the same.
- Since both 'upper' halves have the same area, the places where one path lies inside and the other outside must cancel out.
- The two paths cross, carving those in-between slivers into A (top), B (middle) and C (bottom), with A and C on one side of the balance and B on the other.
- Cancelling gives \(A + C = B\), so the correct statement is B = A + C (answer B).
Mark:
· log in to save