Problem 21 · 2024 Math Kangaroo
Stretch
Geometry & Measurement
spatial-reasoning
A three-sided pyramid has edges with side lengths 5, 6, 7, 8, 9 and 10. The points M, N, P, Q, R and S are the midpoints of the edges, as shown in the diagram. What is the total length of the closed polyline MNPQRSM?
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Answer: C — 21
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Hint 1 of 3
Each step of the polyline joins the midpoints of two edges that meet at a vertex — that is a midsegment of a triangular face.
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Hint 2 of 3
A midsegment is parallel to and exactly half of the third edge of that face.
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Hint 3 of 3
So each of the six steps is half of one edge; add the six halves the closed path uses.
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Approach: every step is a midsegment, so half an edge
- M, N, P, Q, R, S are the six edge midpoints, and each consecutive pair lies on two edges sharing a vertex.
- The segment between two such midpoints is a triangle midsegment, equal to half of the third edge of that face, so each step is half of one edge.
- Reading the faces the path crosses, the six steps are the halves of edges \(10, 5, 6, 7, 8, 6\), giving \(\tfrac{1}{2}(10+5+6+7+8+6)=\tfrac{1}{2}\cdot 42=\) 21 (answer C).
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