In-AB-C-m-b-2-m-c-2-5-m-a-2-prove-that-A-90-m-a-median-

Question Number 169101 by mnjuly1970 last updated on 24/Apr/22

I n A \overset{Δ}{B C} : m_{b}^{2} + m_{c}^{2} = 5 m_{a}^{2}

p r o v e t h a t : \overset{\land}{A} = 90^{°}

m_{a} : (m e d i a n)

Answered by mr W last updated on 24/Apr/22

m_{a} = \frac{\sqrt{2 b^{2} + 2 c^{2} - a^{2}}}{2}

m_{b} = \frac{\sqrt{2 c^{2} + 2 a^{2} - b^{2}}}{2}

m_{c} = \frac{\sqrt{2 a^{2} + 2 b^{2} - c^{2}}}{2}

m_{b}^{2} + m_{c}^{2} = \frac{2 c^{2} + 2 a^{2} - b^{2} + 2 a^{2} + 2 b^{2} - c^{2}}{4}

m_{b}^{2} + m_{c}^{2} = \frac{4 a^{2} + b^{2} + c^{2}}{4}

5 m_{a}^{2} = \frac{5 (2 b^{2} + 2 c^{2} - a^{2})}{4} = \frac{4 a^{2} + b^{2} + c^{2}}{4}

10 b^{2} + 10 c^{2} - 5 a^{2} = 4 a^{2} + b^{2} + c^{2}

b^{2} + c^{2} = a^{2}

\Rightarrow r i g h t a n g l e d t r i a n g l e w i t h ∠ A = 90 °

Commented by mnjuly1970 last updated on 24/Apr/22

t h a n k s a l o t s i r W \dots .