How long is MN?
A hard-level "Length" problem."Join the midpoints of two sides and you get half the third side." Use it twice on a trapezium and the average of the two parallel sides falls out.
GivenAD is parallel to BC. AD = 6 cm, BC = 14 cm. M is the midpoint of AB and N is the midpoint of DC.
Hints
- There is no direct handle on MN. Draw one diagonal, AC, and cut MN into two parts. Call the midpoint of AC point P.
- Look only at the blue triangle ABC. M is the midpoint of AB and P is the midpoint of AC, so MP joins the midpoints of two sides.
- The small red triangle AMP and the blue triangle ABC are the same shape, with every side halved. So MP = BC ÷ 2 = 14 ÷ 2 = 7 cm.
- Do the same on the other side, in triangle ACD. P is the midpoint of AC and N is the midpoint of DC, so PN = AD ÷ 2 = 6 ÷ 2 = 3 cm.
- MN is just those two joined: 7 + 3. Rewritten, that is (6 + 14) ÷ 2 — the average of the two parallel sides.
Answer 10cm
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