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Question Number 181711 by Ari last updated on 28/Nov/22

Commented by mr W last updated on 29/Nov/22

$\frac{9}{53 + 9} \times 48 = \frac{216}{31}$ $\frac{9}{53 - 9} \times 48 = \frac{108}{11}$ $\frac{1}{2} (\frac{216}{31} + \frac{108}{11}) = \frac{2862}{341} \approx 8.4 < 16$ $\Rightarrow t r i a n g l e w i t h h e i g h t 16 i m p o s s i l e!$ $\Rightarrow q u e s t i o n i s w r o n g!$
Commented by Ari last updated on 29/Nov/22

Commented by mr W last updated on 29/Nov/22

Commented by mr W last updated on 29/Nov/22

$M e t h o d I$ $s a y t h e t w o u n k n o w n s i d e s a r e x, y .$ $\frac{x}{y} = \frac{9}{53} . . . (i)$ $\sqrt{x^{2} - 16^{2}} + \sqrt{y^{2} - 16^{2}} = 48 . . . (i i)$ $w e g e t f r o m (i) y = \frac{53 x}{9} a n d p u t t h i s$ $i n t o (i i) . w e g e t a q u a d r a t i c e q n . f o r$ $x^{2} . {w e}^{'} l l s e e i t h a s n o t r e a l r o o t s .$
Commented by mr W last updated on 29/Nov/22

Commented by mr W last updated on 29/Nov/22

$M e t h o d I I$ $B C = 48$ $f o r \frac{B A}{C A} = \frac{9}{53} = c o n s t a n t, t h e l o c u s o f$ $p o i n t A i s a c i r c l e a s s h o w n .$ $\frac{B D}{C D} = \frac{9}{53}$ $\Rightarrow B D = \frac{9}{53 + 9} \times B C = \frac{9}{53 + 9} \times 48 = \frac{216}{31}$ $\frac{B E}{C E} = \frac{9}{53}$ $\Rightarrow B E = \frac{9}{53 - 9} \times B C = \frac{9}{53 - 9} \times 48 = \frac{108}{11}$ $O B = O D = r a d i u s = \frac{B D + B E}{2}$ $= \frac{1}{2} (\frac{216}{31} + \frac{108}{11}) = \frac{2862}{341} \approx 8.39$ $t h e l a r g e s t p o s s i b l e h e i g h t o f t h e$ $t r i a n g l e i s {O A}^{'} = r a d i u s \approx 8.39 < 16 .$ $t h a t m e a n s t h e t r i a n g l e c a n n e v e r$ $h a v e a h e i g h t o f 16 .$
Commented by Ari last updated on 29/Nov/22

$T h a n k y o u v e r y m u c h f o r t h e f u l l c l a r i f i c a t i o n M r . W!$
Commented by mr W last updated on 30/Nov/22

$s e e a l s o Q 181759$
Commented by mr W last updated on 30/Nov/22

$k = \frac{9}{53}$ $r = \frac{a}{\frac{1}{k} - k} = \frac{48}{\frac{53}{9} - \frac{9}{53}} = \frac{2862}{341} \approx 8.39$
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