Question

If a>b>c>0 such taht

p_1 = \sqrt{(a+b)^2+c^2},

p_2= \sqrt{(a+c)^2+b^2},

p_3 =\sqrt{(b+c)^2+a^2},

Compare the size of p_1, p_2 and p_3

Collected in the board: Square Root

Steven Zheng posted 5 hours ago

Answer

\sqrt{(a+b)^2+c^2}

=\sqrt{a^2+b^2+c^2+2ab}

=\sqrt{a^2+b^2+c^2+2abc\cdot\dfrac{1}{c}}
(1)

\sqrt{(a+c)^2+b^2}

=\sqrt{a^2+b^2+c^2+2abc\cdot\dfrac{1}{b}}
(2)

\sqrt{(b+c)^2+a^2}

=\sqrt{a^2+b^2+c^2+2abc\cdot\dfrac{1}{a}}
(3)

\therefore p_1>p_2>p_3


Steven Zheng posted 5 hours ago

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