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Question Number 14965 by b.e.h.i.8.3.4.1.7@gmail.com last updated on 06/Jun/17

Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 06/Jun/17

$i n t r i a n g l e A B D :$ $C F, I H, G J, a r e t h e p e r p e n d i c u l a r$ $b i s e c t o r o f s i d e s .$ $A D = 12, A B = 14, B D = 16$ $. . . . . . . . . . . . . . . \sqrt{. . . . . . .} . . . . === \sqrt{. . . . . . . .} . . . = . . . . .$ $S_{I \overset{Δ}{K J}} = ? (S = a r e a)$
Commented by ajfour last updated on 06/Jun/17

$I J = (K C) (\tan A + \tan B) .$
	Answered by ajfour last updated on 06/Jun/17

	$(r e f e r t o d i a g r a m c o m m e n t e d b e l o w)$ $T o l o c a t e D (x, y) f o r e m o s t :$ $x^{2} + y^{2} = b^{2} = 144$ ${(x - 14)}^{2} + y^{2} = a^{2} = 256$ $s u b t r a c t i n g w e g e t :$ $28 x - 196 = 144 - 256$ $\Rightarrow x = \frac{84}{28} = 3$ $y = \sqrt{b^{2} - x^{2}} = \sqrt{144 - 9} = 3 \sqrt{15}$ $A J = \frac{(b / 2)}{\cos A} = \frac{(b / 2)}{(x / b)} = \frac{b^{2}}{2 x} = \frac{144}{2 \times 3} = 24$ $C J = A J - A C = A J - \frac{d}{2} = 24 - 7 = 17$ $I B = \frac{(a / 2)}{\cos B} = \frac{(a / 2)}{(d - x) / a} = \frac{a^{2}}{2 (d - x)}$ $= \frac{256}{2 (14 - 3)} = \frac{128}{11}$ $I C = I B - B C = I B - \frac{d}{2} = \frac{128}{11} - 7 = \frac{51}{11}$ $I J = I C + C J = \frac{51}{11} + 17 = \frac{238}{11}$ $∠ C K J = ∠ A$ $s o, \tan A = \tan (∠ C K J) = \frac{C J}{C K} = \frac{C J}{h}$ $\Rightarrow h = \frac{C J}{\tan A} = \frac{C J}{(y / x)} = \frac{17}{\sqrt{15}}$ ${A r e a}_{△ I J K} = \frac{1}{2} (I J) h$ $= \frac{1}{2} (\frac{238}{11}) (\frac{17}{\sqrt{15}}) = \frac{119 \times 17}{11 \sqrt{15}}$ $= \frac{7 \times 17 \times 17}{11 \sqrt{15}} \approx 47.5 s q u n i t s ∙$
	Commented by ajfour last updated on 06/Jun/17

	Commented by ajfour last updated on 06/Jun/17

	$n o t t o s c a l e, a s A J > A B w h e n$ $v a l u e s o f a = 16, b = 12, a n d d = 14 .$
	Commented by RasheedSoomro last updated on 06/Jun/17

	$N! CE a p p r o a c h!$ $W ithout analytic approach it will be$ $more challenging!$
	Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 06/Jun/17

	$I H = \frac{2 a . S}{a^{2} + c^{2} - b^{2}} = \frac{S}{c . c o s B}, G J = \frac{S}{c . c o s A}$ $∠ C I K = 90 - ∠ B, ∠ C J K = 90 - ∠ A$ $\frac{G J}{s i n A} = A J \Rightarrow A J = \frac{S}{c . s i n A . c o s A} = \frac{b}{2 c o s A}$ $\Rightarrow \frac{c}{2} + C J = \frac{b}{2 c o s A} = \frac{c . c o s A + a . c o s D}{2 c o s A}$ $\Rightarrow C J = \frac{a . c o s D}{2 c o s A}$ $B I = \frac{I H}{s i n B} \Rightarrow B I = \frac{S}{c . c o s B . s i n B} = \frac{a}{2 c o s B}$ $\Rightarrow \frac{c}{2} + C I = \frac{a}{2 c o s B} = \frac{c . c o s B + b . c o s D}{2 c o s B}$ $\Rightarrow C I = \frac{b . c o s D}{2 c o s B}$ $\Rightarrow I J = C J + C I = \frac{a c o s D}{2 c o s A} + \frac{b c o s D}{2 c o s B} =$ $= \frac{c o s D}{2} (\frac{a c o s B + b c o s A}{c o s A c o s B}) = \frac{c . c o s D}{2 c o s A . c o s B}$ $t g (90 - A) = \frac{C K}{C J} \Rightarrow C K = \frac{c o s A}{s i n A} . \frac{a c o s D}{2 c o s A} =$ $= \frac{a . c o s D}{2 s i n A} \Rightarrow C K = \frac{a . c o s D}{2 s i n A}$ $S_{I \overset{Δ}{K J}} = \frac{1}{2} . C K . I J = \frac{1}{2} . \frac{a c o s D}{2 s i n A} . \frac{c . c o s D}{2 c o s A c o s B}$ $\Rightarrow S_{Δ I K J} = \frac{1}{8} . \frac{a c . {c o s}^{2} D}{s i n A . c o s A . c o s B} =$ $= \frac{{c o s}^{2} D}{4 s i n A c o s A s i n B c o s B} . S = \frac{1 + c o s 2 D}{2 s i n 2 A . s i n 2 B} . S$ $\Rightarrow S_{I J K} = \frac{1 + c o s 2 D}{2 s i n 2 A . s i n 2 B} . S_{A B D}$
	Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 06/Jun/17

	$f o r : a = 16, b = 12, c = 14$ $p = \frac{16 + 12 + 14}{2} = 21$ $S = \sqrt{21 \times 7 \times 9 \times 5} = 21 \sqrt{15}$ $c o s A = \frac{144 + 196 - 256}{2 \times 12 \times 14} = \frac{1}{4}, s i n A = \frac{\sqrt{15}}{4}$ $c o s B = \frac{256 + 196 - 144}{2 \times 16 \times 14} = . 68, s i n B = . 73$ $c o s D = \frac{256 + 144 - 196}{2 \times 12 \times 16} = . 53$ $s i n 2 A = 2 \times \frac{1}{4} \times \frac{\sqrt{15}}{4} = . 48$ $s i n 2 B = 2 \times . 68 \times . 73 = . 99$ $c o s 2 D = 2 \times {(. 53)}^{2} - 1 = - . 44$ $\Rightarrow S_{I J K} = \frac{1 - . 44}{2 \times . 48 \times . 99} . 21 \sqrt{15} = 48.07 . ◼$
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