The Law of Cosines

The Law of Cosines

The Law of Cosines is the relationship between sides and angles in any triangle.

Introduction

Let's prove the theorem.
  • Draw a triangle ABC,then draw three squares sharing their sides with the triangle ABC.
  • Draw three lines through A,B,and C so that each line is perpendicular to the opposite side.
  • You can prove A1=A2, A3=A4, and A5=A6 by using the applet bellow.
  • Now you understand
    a^2=A2+A3=A1+A4=(c^2-A6)+(b^2-A5)
    =c^2+b^2-2*A6
  • Find the expression for the area A6.

How to use the applet.

The applet

How to use the applet.
  • Select a rectangle from "green", "pink", and "blue".
  • Select the transformation from "shift", "rotate", and "move".
  • Drag the red points to transform the rectangle.
Activity
  1. Transform A1 to A2.
  2. Transform A3 to A4.
  3. Transform A5 to A6.
Mr. KOBAYASHI Akira at Seisan Circle(a math education research group) told us the idea of the applet.