April 4th, 2010, 10:32 AM  #1 
Newbie Joined: Mar 2010 Posts: 9 Thanks: 0  Analytic geometry help please
Help please, Calculate the area of the triangle with corners A = (3, 2), B = (2, 1) and C = (1, 3) I know what to do but I'm not quite sure how to; I want to caculate the lengths between ab, bc using the formula square root of (y2y1)^2+(x2x1)^2. So: ab: sqroot ((1+2)^2+(23)^2) = sqroot (9+1) = sqroot (10) bc: sqroot ((31)^2+(12)^2) = sqroot (4+9) = sqroot (13) ac: sqroot ((3+2)^2+(13)^2) = sqroot (25+16) = sqroot (41) (I'm not sure if these are correct) since it's neither a right triangle nor are the feet the same length I can't just use Pythagoras or take base/2 and then pythagoras... so I don't really know what to do now.. I don't even know if I started out right on this because it seems a tad complicated to be the third task in the first practisetest of the course in the book (the first ones were a breeze) 
April 4th, 2010, 02:49 PM  #2 
Senior Member Joined: Apr 2008 Posts: 435 Thanks: 0  Re: Analytic geometry help please
There are several ways to proceed, and I can't begin to guess which ones you would know. The easiest way, which I can only hope is what you've learnt, would be to use Heron's formula. Or, you could note that if you split a triangle into two right triangles, you can come up with a formula (which happens to be Heron's formula), as well. I don't really know if that's helpful or not. 
April 4th, 2010, 03:20 PM  #3 
Senior Member Joined: Feb 2010 Posts: 711 Thanks: 147  Re: Analytic geometry help please
If this is a question from a standard analysis or precalculus course then you might want to look in your notes or your book for a determinant form. This will allow you to find the area from the coordinates of the vertices by just evaluating a particular 3 x 3 determinant. MrTwhs 
April 4th, 2010, 03:36 PM  #4  
Senior Member Joined: Apr 2008 Posts: 435 Thanks: 0  Re: Analytic geometry help please Quote:
 
April 4th, 2010, 06:44 PM  #5 
Senior Member Joined: Feb 2010 Posts: 711 Thanks: 147  Re: Analytic geometry help please
If the vertices of the triangle are then the area of the triangle is given by Area = where the sign is determined by the direction in which the vertices are traversed. 
April 4th, 2010, 09:29 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,931 Thanks: 2207 
Let O = (0, 0) and D = (7, 0). OC, AB are parallel, so triangles ABC, ABO have the same area. OB, AD are parallel, so triangles ABO, DBO have the same area. In triangle DBO, choose OD (which has length 7) as base, so that the corresponding height is 1. Area triangle ABC = area triangle DBO = base × height / 2 = 7 × 1 / 2 = 3.5. 
April 5th, 2010, 01:07 AM  #7  
Senior Member Joined: Apr 2008 Posts: 435 Thanks: 0  Re: Analytic geometry help please Quote:
 

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