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 October 3rd, 2012, 04:15 PM #1 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,414 Thanks: 1024 Right triangle area Right triangle ABC, the 2 legs being a and b. Let radius of incircle = r Show that area of ABC = (a - r)(b - r) Kinda fun!
 October 3rd, 2012, 06:08 PM #2 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,935 Thanks: 1129 Math Focus: Elementary mathematics and beyond Re: Right triangle area $\text{Let }A\text{ be the area of the triangle.}$ $A\,=\,2\,\cdot\,\frac{a\,-\,r}{2}\,\cdot\,r\,+\,2\,\cdot\,\frac{b\,-\,r}{2}\,\cdot\,r\,+\,r^2 \\ =\,(a\,-\,r)\,\cdot\,r\,+\,(b\,-\,r)\,\cdot\,r\,+\,r^2 \\ =\,r(a\,+\,b\,-\,r)$ $(a\,-\,r)(b\,-\,r)\,=\,ab\,-\,ar\,-\,br\,+\,r^2\,=\,ab\,-\,r(a\,+\,b\,-\,r)\,=\,ab\,-\,\frac{ab}{2}\,=\,\frac{ab}{2}$ $\text{Q. E. D.}$
October 3rd, 2012, 07:07 PM   #3
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Re: Right triangle area

[attachment=0:14dz3n3u]Denistriangle.jpg[/attachment:14dz3n3u]

By Pythagoras:

$a^2+b^2=c^2=(a+b-2r)^2=a^2+b^2+2ab-4ar-4br+4r^2$

$ab-ar-br+r^2=ar+br-r^2$

$(a-r)(b-r)=r(a-r)+r(b-r)+r^2=A$
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 Denistriangle.jpg (10.1 KB, 334 views)

 October 3rd, 2012, 07:09 PM #4 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,414 Thanks: 1024 Re: Right triangle area Nice one Greg. I went this way: (a - r) (b - r) = area? Since r = (a + b - c) / 2 : [a - (a + b - c) / 2] * [b - (a + b - c) / 2] = area Expand and simplify: (c^2 - a^2 - b^2 + 2ab) / 4 = area Since c^2 - a^2 - b^2 = 0, then: 2ab / 4 = area ab / 2 = area Anything "wrong"? Doesn't "feel" 100% ok !
 October 4th, 2012, 02:19 AM #5 Global Moderator   Joined: Dec 2006 Posts: 20,629 Thanks: 2077 Nothing is wrong, but there is no need to repeat "= area" until the end. (a - r)(b - r) = (a - (a + b - c)/2)(b - (a + b - c)/2)                   = (c + a - b)(c - a + b)/4                   = (c² - (a - b)²)/4                   = (c² - (a² + b²) + 2ab)/4                   = ab/2                   = area of triangle ABC.
 October 4th, 2012, 03:40 AM #6 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,414 Thanks: 1024 Re: Right triangle area OOOOOOOOOOOOOOOOOOOKKKKKKKKKKKKKKKKKKKKKKK! Thanks, Skip.

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