Red-Area-

Question Number 198210 by a.lgnaoui last updated on 14/Oct/23

Red Area ?

Commented by a.lgnaoui last updated on 14/Oct/23

Answered by mr W last updated on 14/Oct/23

Commented by mr W last updated on 14/Oct/23

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

A = \frac{a b}{2}

\frac{A_{1}}{A} = {(\frac{c}{a})}^{2} \Rightarrow A_{1} = \frac{c^{2}}{a^{2}} A

\frac{x}{c} = \frac{b}{a} \Rightarrow x = \frac{b c}{a}

y = c - x = c - \frac{b c}{a} = (1 - \frac{b}{a}) c

\frac{A_{2}}{A} = {(\frac{y}{a})}^{2} \Rightarrow A_{2} = {(1 - \frac{b}{a})}^{2} \frac{c^{2}}{a^{2}} A

A_{1} + A_{2} = [1 + {(1 - \frac{b}{a})}^{2}] \frac{a^{2} + b^{2}}{a^{2}} \times \frac{a b}{2}

A_{1} + A_{2} = \frac{(a^{2} + b^{2}) (2 a^{2} + b^{2} - 2 a b) b}{2 a^{3}}

A_{s h a d e d} = c^{2} - A_{1} - A_{2} = a^{2} + b^{2} - \frac{(a^{2} + b^{2}) (2 a^{2} + b^{2} - 2 a b) b}{2 a^{3}}

A_{s h a d e d} = \frac{(a^{2} + b^{2}) (2 a^{3} - 2 a^{2} b + 2 {a b}^{2} - b^{3})}{2 a^{3}} ✓

Commented by a.lgnaoui last updated on 14/Oct/23

thanks