Inequalities for Geometric Problems Practice

Answer 1

1. In a triangle ABC, the sum of the lengths of any two sides must be greater than the length of the third side.
AC + BC > AB
AB + BC > AC
AB + AC > BC

2. In a right triangle, the length of the hypotenuse (c) is always greater than the lengths of the other two sides (a and b).
c > a
c > b

3. In a quadrilateral, the sum of any three sides must be greater than the length of the fourth side.
AB + BC + CD > AD
AB + AC + AD > BC
AB + BD + CD > AC

4. In a circle, the circumference of the circle must be greater than the sum of the lengths of any two sides of a triangle inscribed in the circle.
2πr > AB + BC
2πr > AC + BC
2πr > AB + AC

5. In any polygon, the sum of the lengths of any two sides must be greater than the length of the diagonal connecting those two sides.
AD > AB + BD
BE > BC + CD
CF > AC + BC

6. In a rectangle, the sum of the lengths of any two sides must be greater than the length of the diagonal.
AC > AB + BC
BD > AB + AD

7. In a square, the perimeter of the square must be greater than the length of the diagonal.
4a > √2a