Math Question and Answers


Question and Answers Forum

All Questions Topic List
Mensuration Questions
Previous in All Question Next in All Question
Previous in Mensuration Next in Mensuration
Question Number 158740 by ajfour last updated on 08/Nov/21

Commented by ajfour last updated on 08/Nov/21

$F i n d a r e a o f △ A B C i n$ $t e r m s o f a, b, c .$
	Answered by ajfour last updated on 08/Nov/21
	$2△=ah (√(c^2 −h^2 ))+(√(b^2 −h^2 ))=a ⇒ (c^2 +b^2 −a^2 −2h^2 )^2 =4{h^4 −(b^2 +c^2 )h^2 +b^2 c^2 } ⇒ (c^2 +b^2 −a^2 )^2 −4h^2 (c^2 +b^2 −a^2 ) +4(b^2 +c^2 )h^2 −4b^2 c^2 =0 4a^2 h^2 ={(b+c)^2 −a^2 }{a^2 −(b−c)^2 } =16△^2 (2△)^2 =b^2 c^2 −(1/4)(c^2 +b^2 −a^2 )^2 △=(1/4)(√((b+c)^2 −a^2 ))(√(a^2 −(b−c)^2 ))$
	$2 △ = a h$ $\sqrt{c^{2} - h^{2}} + \sqrt{b^{2} - h^{2}} = a$ $\Rightarrow {(c^{2} + b^{2} - a^{2} - 2 h^{2})}^{2}$ $= 4 {h^{4} - (b^{2} + c^{2}) h^{2} + b^{2} c^{2}}$ $\Rightarrow {(c^{2} + b^{2} - a^{2})}^{2} - 4 h^{2} (c^{2} + b^{2} - a^{2})$ $+ 4 (b^{2} + c^{2}) h^{2} - 4 b^{2} c^{2} = 0$ $4 a^{2} h^{2} = {{(b + c)}^{2} - a^{2}} {a^{2} - {(b - c)}^{2}}$ $= 16 △^{2}$ ${(2 △)}^{2} = b^{2} c^{2} - \frac{1}{4} {(c^{2} + b^{2} - a^{2})}^{2}$ $△ = \frac{1}{4} \sqrt{{(b + c)}^{2} - a^{2}} \sqrt{a^{2} - {(b - c)}^{2}}$
	Commented by mr W last updated on 08/Nov/21

	$f u r t h e r$ $△ = \frac{1}{4} \sqrt{(a + b + c) (- a + b + c) (a - b + c) (a + b - c)}$ $t h i s i s t h e {h e r o n}^{'} s f o r m u l a .$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum