## 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, (`a``c`)(`d``b`)/2 = (`ad``ab``cd`+`bc`)/2, and `ab`/2. Therefore the total area of these triangles is (`ad`+`bc`)/2.

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

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

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