Revision Notes: Inequalities

Important Theorems 

• If two sides of a triangle are unequal, the greater side has the greater angle opposite to it. 
• If two angles of a triangle are unequal, the greater angle has the greater side opposite to it. 
• The sum of any two sides of a triangle is greater than the third side. 
• In a triangle, the difference of the lengths of any two sides is less than the third side. 
• Of all the straight lines that can be drawn to a straight line from a given point outside it, the perpendicular is the shortest. 
• Perimeter of a triangle is greater than the sum of the lengths of its medians.
  Thus, we have, AB + BC + AC > AD + BE + CF
• In any quadrilateral, the sum of the lengths of four sides exceeds the sum of the lengths of its diagonals. 
  Thus, we have (AB + BC + CD + D) A > (AC + BD)
• If P is any point in the interior of a triangle ABC, then we have
  PA + PB < AC + BC