M 340L HW 8 Solutions

1 hint translate abc so that a becomes the origin

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Unformatted text preview: 0.0 points Show that the triangle ∆ABC with vertices at A = (a1 , a2 ), B = (b1 , b2 ), C = (c1 , c2 ) has a1 1 b1 det Area = 2 c1 a2 b2 c2 1 1 . 1 (Hint: translate ∆ABC so that A becomes the origin.) Explanation: After translating ∆ABC so that A becomes the origin, we obtain a new triangle ∆OB ′ C ′ of equal area with vertices at the origin and B ′ = (b1 − a1 , b2 − a2 ), C ′ = (c1 − a1 , c2 − a2 ). Now Area(∆OB ′ C ′ ) = 1 b − a1 det 1 c1 − a1 2 b2 − a 2 c2 − a2 . On the other hand, by properties of determinants, a1 a2 1 det b1 b2 1 c1 c2 1 a1 a2 1 = det b1 − a1 b2 − a − 2 0 c1 c2 1 a1 a2 1 = det b1 − a1 b2 − a − 2 0 c1 − a1 c2 − a2 0 = det b1 − a 1 c1 − a1 b2 − a 2 c2 − a2 Consequently, ∆ABC has a1 a2 1 b1 b2 det Area = 2 c1 c2 1 1 1 . . 5...
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This document was uploaded on 03/16/2014 for the course M 340L at University of Texas at Austin.

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