Question-206937

Question Number 206937 by efronzo1 last updated on 01/May/24

Answered by A5T last updated on 01/May/24

Commented by A5T last updated on 01/May/24

8 \times 14 = x (x + 2 r) = x^{2} + 2 r x = 112

14^{2} = {(x + r)}^{2} + r^{2} = x^{2} + 2 r x + 2 r^{2} = 112 + 2 r^{2}

\Rightarrow r^{2} = 42 \Rightarrow r = \sqrt{42} \Rightarrow x = \sqrt{154} - \sqrt{42}

s i n θ = \frac{\sqrt{42}}{14} \Rightarrow c o s θ = \frac{\sqrt{154}}{14}

{A B}^{2} = 8^{2} + {(\sqrt{154} - \sqrt{42})}^{2} - 16 (\sqrt{154} - \sqrt{42}) \frac{\sqrt{154}}{14}

\Rightarrow A B = 2 \sqrt{21 - 3 \sqrt{33}}

{B C}^{2} = 4 r^{2} = {A B}^{2} + {A C}^{2} \Rightarrow A C = 2 \sqrt{21 + 3 \sqrt{33}}

\Rightarrow \frac{A B \times A C}{2} = 2 \sqrt{21^{2} - 9 \times 33} = 2 \times 12 = 24

Commented by A5T last updated on 01/May/24

A f t e r f i n d i n g r, o n e c a n a l s o d o t h i s :

L e t p e r p e n d i c u l a r f r o m A m e e t B C a t D

T h e n, \frac{A D}{r} = \frac{8}{14} \Rightarrow A D = \frac{4 r}{7}

[A B C] = \frac{A D \times B C}{2} = \frac{4 r^{2}}{7} = 24

Answered by mr W last updated on 01/May/24

Commented by mr W last updated on 01/May/24

\frac{\frac{6}{2}}{r} = \frac{r}{8 + 6} \Rightarrow r^{2} = 42

\frac{h}{r} = \frac{8}{8 + 6} = \frac{4}{7}

s h a d e d a r e a = \frac{2 r h}{2} = \frac{4 r^{2}}{7} = \frac{4 \times 42}{7} = 24

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