Question

How are the proofs for the side length ratios of 30-60-90 and 45-45-90 triangles similar? How are they different?

Answer 1

As the ratios are different hence the two triangles are not similar.

Explanation:

The side lengths ratios of one triangle are given as:

30-60-90

i.e. the sides are of length: 30x,60x,90x for some real number 'x'

and the side length ratio of the other triangle is given as:

45-45-90

i.e. the sides of the second triangle is given by:

45y,45y,90y for some real number 'y'.

" If all the sides of a triangle are proportional to the corresponding sides of another triangle then the triangles are said to be similar "

Now we check the ratio as:

• 
  `(30x)/(45y)=(2x)/(3y)`

• 
  `(60x)/(45y)=(4x)/(3y)`

• 
  `(90x)/(90y)=(x)/(y)`

As the 3 ratios are not equal hence the two triangles are not similar.