Question

The lengths of two line segments are $a$ units and $b$ units respectively. Then the correct relation between them is:

$\dfrac{a+b}{2}\gt\sqrt{ab}$

$\dfrac{a+b}{2}\lt\sqrt{ab}$

$\dfrac{a+b}{2}=\sqrt{ab}$

$\dfrac{a+b}{2}\le\sqrt{ab}$

$\dfrac{a+b}{2}\ge\sqrt{ab}$

Start with the nonnegative square $(a-b)^2$

Equality must remain possible when the two segment lengths are equal

Since $(a-b)^2\ge0,$ we have $(a+b)^2\ge4ab.$ Both sides are nonnegative, so $\frac{a+b}{2}\ge\sqrt{ab}.$ Equality occurs when $a=b.$

Thus, the correct answer is E.