Connect the Midpoints — Exactly Half
Join the midpoints of a rectangle to form a rhombus. No matter the proportions, the rhombus is always exactly half. Why?
Do not write an equation yet. Shake the figure first.
Flatten the rectangle until it looks wrong
The rhombus changes shape, while the midpoint area ratio remains exactly 50%.
Play the reversal slowly
Now move the same pieces step by step and prove that the pattern you touched was not an accident.
Join the four midpoints into a rhombus. What is its area?
Return to the exact question
Joining the four side-midpoints of a rectangle forms a rhombus. What fraction of the rectangle is the rhombus?
Why this approach works
Don't compute the ratio — watch the leftover pieces. The four corner right-triangles each rotate 180° about a side-midpoint, and all four apexes land at the rectangle's center, exactly tiling the rhombus. So the four outer pieces equal the inner rhombus; together they make the whole rectangle. The rhombus is exactly half — independent of shape.
Proof
The four corner triangles rotate to tile the rhombus exactly, so the rhombus equals half the rectangle — independent of proportions.
Try it yourself
Imagine a very flat rectangle, or a square. The corner triangles change shape, but four of them always tile the rhombus — the ratio never moves off ½.
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