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Question Number 154910 by mr W last updated on 23/Sep/21

Commented by mr W last updated on 23/Sep/21

$t e t r a h e d r o n T - A B C h a s e d g e l e n g t h e s$ $a, b, c, u, v, w .$ $t h e f a c e s o f t e t r a h e d r o n T^{'} - A^{'} B^{'} C^{'}$ $h a v e t h e d i s t a n c e s d_{1}, d_{2}, d_{3}, d_{4} t o t h e$ $c o r r e s p o n d i n g f a c e s o f T - A B C .$ $f i n d t h e v o l u m e o f n e w t e t r a h e d r o n .$
Commented by Rasheed.Sindhi last updated on 23/Sep/21

$G u e s s$ $V^{'} = {(1 + \frac{(\overset{a r e a}{f a c e 1}) d_{1} + (\overset{a r e a}{f a c e 2}) d_{2} + (\overset{a r e a}{f a c e 3}) d_{3} + (\overset{a r e a}{f a c e 4}) d_{4}}{3 V})}^{3} . V$
	Answered by mr W last updated on 24/Sep/21

	$t e t r a h e d r o n T - A B C :$ $e d g e s : a, b, c, u, v, w$ $v o l u m e : V i n t e r m s o f a, b, c, u, v, w^{)}$ $f a c e 1 : Δ T B C, a r e a A_{1} i n t e r m s o f a, v, w^{ )}$ $f a c e 2 : Δ T C A, a r e a A_{2} i n t e r m s o f b, w, u$ $f a c e 3 : Δ T A B, a r e a A_{3} i n t e r m s o f c, u, v$ $f a c e 4 : Δ A B C, a r e a A_{4} i n t e r m s o f a, b, c$ $r a d i u s o f i n s p h e r e o f T - A B C : r$ $\frac{r (A_{1} + A_{2} + A_{3} + A_{4})}{3} = V$ $t e t r a h e d r o n s T - A B C a n d T^{'} - A^{'} B^{'} C^{'}$ $a r e s i m i l a r .$ $s a y t h e e d g e s o f T^{'} - A^{'} B^{'} C^{'}$ $a r e a^{'}, b^{'}, c^{'}, u^{'}, v^{'}, w^{'} a n d$ $a^{'} = k a, b^{'} = k b, c^{'} = k c, u^{'} = k u, v^{'} = k v, w^{'} = k w$ $w i t h k = f a c t o r o f m a g n i f i c a t i o n .$ $w e h a v e$ $a r e a o f f a c e 1 o f T^{'} - A^{'} B^{'} C^{'} A_{1}^{'} = k^{2} A_{1}$ $a r e a o f f a c e 2 o f T^{'} - A^{'} B^{'} C^{'} A_{2}^{'} = k^{2} A_{2}$ $a r e a o f f a c e 3 o f T^{'} - A^{'} B^{'} C^{'} A_{3}^{'} = k^{2} A_{3}$ $a r e a o f f a c e 4 o f T^{'} - A^{'} B^{'} C^{'} A_{4}^{'} = k^{2} A_{4}$ $v o l u m e o f T^{'} - A^{'} B^{'} C^{'} V^{'} = k^{3} V$ $o n t h e o t h e r s i d e,$ $V^{'} = \frac{A_{1}^{'} (r + d_{1})}{3} + \frac{A_{2}^{'} (r + d_{2})}{3} + \frac{A_{3}^{'} (r + d_{3})}{3} + \frac{A_{4}^{'} (r + d_{4})}{3}$ $V^{'} = \frac{k^{2} A_{1} (r + d_{1})}{3} + \frac{k^{2} A_{2} (r + d_{2})}{3} + \frac{k^{2} A_{3} (r + d_{3})}{3} + \frac{k^{2} A_{4} (r + d_{4})}{3}$ $V^{'} = \frac{k^{2} [(A_{1} + A_{2} + A_{3} + A_{4}) r + A_{1} d_{1} + A_{2} d_{2} + A_{3} d_{3} + A_{4} d_{4}]}{3}$ $V^{'} = \frac{k^{2} (3 V + A_{1} d_{1} + A_{2} d_{2} + A_{3} d_{3} + A_{4} d_{4})}{3}$ $\frac{k^{2} (3 V + A_{1} d_{1} + A_{2} d_{2} + A_{3} d_{3} + A_{4} d_{4})}{3} = k^{3} V$ $\Rightarrow k = 1 + \frac{A_{1} d_{1} + A_{2} d_{2} + A_{3} d_{3} + A_{4} d_{4}}{3 V}$ $v o l u m e o f T^{'} - A^{'} B^{'} C^{'} i s$ $V^{'} = {(1 + \frac{A_{1} d_{1} + A_{2} d_{2} + A_{3} d_{3} + A_{4} d_{4}}{3 V})}^{3} V$ $___________________________$ $^{)} u s i n g {E u l e r}^{'} f o r m u l a, s e e Q 40469$ $^{* *)} u s i n g {H e r o n}^{'} s f o r m u l a$
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