Problem 12 · 2014 Math Kangaroo
Medium
Geometry & Measurement
pythagorean-triplearea-decomposition
Paul hangs rectangular pictures on a wall. For each picture he hammers a nail into the wall 2·5 m above the floor and ties a 2 m long string to the two upper corners of the picture (see diagram). Which picture size (width in cm × height in cm) has its lower edge nearest to the floor?
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Answer: C — 120 × 90
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Hint 1 of 2
The 2 m string and the picture's width form an isosceles triangle hanging from the nail.
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Hint 2 of 2
Find how far the top edge drops below the nail, then subtract the picture's height.
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Approach: drop of the top edge below the nail, then subtract height
- Each half of the 2 m string is 1 m and spans half the width w/2 horizontally.
- The top edge sits √(1 − (w/2)²) below the nail (in metres), and the nail is 2.5 m up.
- Lower edge height = 2.5 − √(1 − (w/2)²) − height; testing the options, 120 × 90 gives the smallest (0.8 m).
- So 120 × 90 hangs lowest.
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