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Calculating Triangular Area and Geometric Examples

This document explores the calculations of triangular areas using vertices coordinates. It includes detailed formulas for determining the areas of triangles formed by various points (A, B, C) in a Cartesian plane. The examples illustrate scenarios where the area is calculated based on given vertices, highlighting the use of the triangle area formula, geometric significance, and conditions for collinearity. Each example is tailored to guide the understanding of geometric principles and area determination in two-dimensional space.

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Calculating Triangular Area and Geometric Examples

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  1. y B(x2 , y2) P R A(x1 , y1) x S C(x3 , y3) 三角形面積 長方形PRSC面積 = (x1 - x3 ) (y2 - y3 ) (x3 , y2) (x1 , y2) DBRA面積= 1/2(x1 - x2 ) (y2 - y1 ) DASC面積= 1/2(x1 - x3 ) (y1 - y3 ) DPBC面積= 1/2(x2 - x3 ) (y2 - y3 ) DABC面積= 1/2(x1 y2 +x2y3 +x3y1 - x2 y1 - x3y2 -x1y3 ) (x1 , y3)

  2. y 逆時針方向 B(x2 , y2) - x1 y3 - x3 y2 - x2 y1 + x1 y2 + x2 y3 + x3 y1 A(x1 , y1) x C(x3 , y3) x1 y1 x2 y2 DABC面積= 1/2 x3 y3 x1 y1 DABC面積= 1/2(x1 y2 +x2y3 +x3y1 - x2 y1 - x3y2 -x1y3 ) =1/2[(x1 y2 - x2 y1)+(x2y3 - x3y2)+ (x3 y1 -x1y3 )] x1 y1 x2 y2 x3 y3 x1 y1

  3. y A(-1, 5) B(-4, 1) x C(2, -1) 例子 找出由頂點 A(-1, 5), B(-4, 1) and C(2, -1).所組成雞三角形面積 DABC面積= = 15平方單位 DABC面積= = -15平方單位

  4. Example 2 己知三點為A(0, -8), B(1, -2) and C(3, k) ，三點均在同一直線上。試找k的值。 幾何意義

  5. 例子3 已知A(0, 4) , B(2, 6)和C(-2, k). 若ABC的面積為4平方單位，試求k的值。 幾何意義

  6. Example 4 已知ABC的頂點為A(3, 2), B(-1, 1)和 C(5, -4). 試找出由A到BC的垂直距離。

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