Problem 20 · 2013 Math Kangaroo
Stretch
Geometry & Measurement
area-fraction
If I join the midpoints of the sides of the large triangle in the picture, a small triangle is formed. If I join the midpoints of the sides of this small triangle, a tiny triangle is formed. How many of these tiny triangles can fit into the largest triangle at the same time?
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Answer: D — 16
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Hint 1 of 3
When you join the midpoints of a triangle, it splits into 4 equal little triangles just like it.
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Hint 2 of 3
So the big triangle holds 4 small triangles, and each small triangle holds 4 tiny ones.
Still stuck? Show hint 3 →
Hint 3 of 3
Count the tiny ones: 4 small triangles, each made of 4 tiny ones.
Show solution
Approach: each midpoint-join splits a triangle into 4 copies, so do it twice
- Joining the midpoints cuts the big triangle into 4 equal small triangles.
- Doing it again cuts each of those small triangles into 4 tiny ones.
- That is 4 groups of 4 tiny triangles, so 4 × 4 = 16 fit in the big triangle, which is answer D.
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