Problem 19 · 2016 Math Kangaroo
Stretch
Geometry & Measurement
perimeter
The perimeter of rectangle ABCD is 30 cm. Three more rectangles are added so that their centres are at the corners A, B and D and their sides are parallel to the rectangle (see diagram). The sum of the perimeters of these three rectangles is 20 cm. What is the length of the border of the whole shape (the thick black line)?
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Answer: C — 40 cm
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Hint 1 of 3
Look at one corner rectangle: its centre is on the corner, so exactly half of it sits outside the big rectangle and half sits inside.
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Hint 2 of 3
Trace the thick line around that corner and compare it to the plain corner it replaced.
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Hint 3 of 3
Each corner rectangle adds only half of its own perimeter to the outline.
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Approach: see how much each corner rectangle adds to the outline
- Because each small rectangle is centred on a corner, the big rectangle's two edges cut it into four equal quarters, so half of the small rectangle pokes outside.
- Tracing the thick line, the bits that poke out add length while the bits tucked inside hide the same length, so each corner rectangle adds exactly half of its own perimeter to the outline.
- The three corner rectangles have perimeters adding to 20 cm, so together they add \(\tfrac{1}{2}\times 20 = 10\) cm.
- The outline is the big rectangle's perimeter plus this, \(30 + 10 = 40\) cm, choice (C).
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