How would you prove that the area of an equilateral triangle constructed on the hypotenuse of a right triangle is equal to the sum of the areas of the equilateral triangles constructed on the legs?

1 Answer
Jul 23, 2018

As proved.

Explanation:

Statement of 'Pythagoras theorem': In a right triangle the area of the square on the hypotenuse is equal to the sum of the areas of the squares of its remaining two sides.

#:. a^2 + b^2 = c^2#

http://www.pbs.org/wgbh/nova/proof/puzzle/theoremsans.html

Now let's replace the a, b, c squares by equilateral triangles of sides a, b, c.

#"Given ; "a^2 + b^2 = c^2, " from the right triangle"#

Area of equilateral triangle on hypotenuse c #A_c = (sqrt3/4) c^2#

Similarly, #area of equilateral triangle on side a #A_a = (sqrt3/4) a^2

Likewise, #area of equilateral triangle on side a #A_b = (sqrt3/4) b^2

#A_a + A_b = (sqrt3/4)a^2 + (sqrt3/4)b^2#

#=> (sqrt3/4) (a^2 + b^2) = (sqrt3/4) c^2#