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Triangle Area

Problem

In the diagram below, find the area of triangle OXY in terms of a, b, c, and d.



Solution

Consider the following diagram.


The area of the surrounding rectangle is ad.
The area of each of the surrounding triangles are given by cd/2, (aminusc)(dminusb)/2 = (adminusabminuscd+bc)/2, and ab/2. Therefore the total area of these triangles is (ad+bc)/2.

Hence the area of triangle OXY is ad minus (ad+bc)/2 = (adminusbc)/2.

Given two general points X (a, b) and Y (c, d), prove that the area of triangle OXY is given by |adminusbc|/2, where |n| represents the absolute value of n.

Problem ID: 239 (10 Aug 2005)     Difficulty: 2 Star

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