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AD is the median from A to side BC in triangle ABC. Which statement is NOT always true?

Class 7The Triangle and its PropertiesTriangles
  • AD is the midpoint of BC.
  • BAD is perpendicular to BC.
  • CBD = DC.
  • DTriangles ABD and ACD have equal areas.

Step-by-Step Solution

A median guarantees only that it ends at the midpoint, so BD = DC and the two halves have equal areas (equal bases, same height measured from A). Perpendicularity is the defining feature of an ALTITUDE, and a median is perpendicular only in special cases such as AB = AC. Trap: merging the two definitions and assuming every median meets the side at 90°.
Answer: B) AD is perpendicular to BC.

Key idea: Triangle inequality

**The sum of any two sides of a triangle is greater than the third side.**

So a triangle with sides 3, 4, 5 works (3+4>5, 3+5>4, 4+5>3). But sides 1, 2, 5 do NOT form a triangle — 1 + 2 = 3 is less than 5, so the two short sides can't reach across.

Use this to test whether three given lengths can be a triangle, or to bound a missing side: if two sides are 5 and 8, the third must be greater than 8 − 5 = 3 AND less than 8 + 5 = 13.

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