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Question Number 9907 by amir last updated on 15/Jan/17

Commented by RasheedSoomro last updated on 15/Jan/17

$What is meant by h_{a} ?$
Commented by amir last updated on 15/Jan/17

$h_{a} i s t h e p e r p e n d i c u l a r l i n e f r o m A p o i n t t o B C s i d e .$
	Answered by mrW1 last updated on 21/Jan/17

	$Δ A B C w i t h s i d e s a, b, c a n d a r e a S .$ $\frac{1}{2} \times a \times h_{a} = S$ $\Rightarrow \frac{1}{h_{a}} = \frac{a}{2 S}$ $\frac{1}{2} \times b \times h_{b} = S$ $\Rightarrow \frac{1}{h_{b}} = \frac{b}{2 S}$ $\frac{1}{2} \times c \times h_{c} = S$ $\Rightarrow \frac{1}{h_{c}} = \frac{c}{2 S}$ $i t m e a n s t h e n e w t r i a n g l e w i t h t h e s i d e s o f$ $\frac{1}{h_{a}}, \frac{1}{h_{b}}, \frac{1}{h_{c}} i s s i m i l a r w i t h t h e o r i g i n a l t r i a n g l e$ $w i t h t h e s i d e s o f a, b, c .$ $t h e r a t i o o f t h e c o r r e s p o n d i n g s i d e s$ $i s \frac{1}{2 S} .$ $t h e r a t i o o f t h e a r e a s o f b o t h t r i a n g l e s$ $i s t h e s q u a r e o f t h e r a t i o o f t h e i r s i d e s .$ $\Rightarrow \frac{S_{1}}{S} = {(\frac{1}{2 S})}^{2} = \frac{1}{4 S^{2}}$ $\Rightarrow S \cdot S_{1} = \frac{1}{4}$
	Commented by amir last updated on 29/Jan/17

	$S a n d S_{1} a r e t h e s a m e q u a n t i t e s .$ $s o h o w c a n t h e i r p r o d u c t e q v a l t o$ $a p u r e n u m b e r ?$ $i t h i n k t h a t t h i s i s i m p o s s i b l e .$ $i t i s a p a r a d o x .$
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