Problem 28 · 2012 Math Kangaroo
Stretch
Geometry & Measurement
taxicab-distance
Three corners of a die (not all on one face) have the coordinates P(3, 4, 1), Q(5, 2, 9) and R(1, 6, 5). What are the coordinates of the midpoint of the die?
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Answer: A — A(4, 3, 5)
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Hint 1 of 2
Compute the squared distances between \(P\), \(Q\), \(R\) and compare them.
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Hint 2 of 2
If two of the points turn out to be opposite corners (a space diagonal), the centre of the die is just their midpoint.
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Approach: identify the space diagonal, then take its midpoint
- The squared distances are \(PR^2 = 24\), \(QR^2 = 48\), \(PQ^2 = 72\), in the ratio \(1 : 2 : 3\).
- For a cube with edge\(^2 = 24\) these are an edge, a face diagonal and a space diagonal, so \(P\) and \(Q\) are opposite corners.
- The centre is the midpoint of \(PQ\): \(\left(\tfrac{3+5}{2}, \tfrac{4+2}{2}, \tfrac{1+9}{2}\right) = (4,3,5)\), choice A.
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