A 30-60-90 triangle...

Question

 

A 30−60−90∘ triangle has side lengths with the ratio 1:2:3. A 45−45−90∘ triangle has side lengths with the ratio 1:1:2.

a. What is the perimeter of a basic 30−60−90∘ triangle (with side lengths equal to the ratio values)? What is the perimeter of a basic 45−45−90∘ triangle? Write your answers in both simplified and estimated form.

b. What is the ratio of the 30−60−90∘ triangle’s perimeter to the 45−45−90∘ triangle’s perimeter? Show how you arrived at your answer, and write your answer in both simplified and estimated form. Is this ratio true for all 30−60−90∘ and 45−45−90∘ triangles?

Answer 1

a 30-60-90 triangle has sides in the ratio of 1:sqr3:2, not 1:2:3

a 45-45-90 triangle has sides in the ratio of 1:1:sqr2, not 1:1:2

perimeter of a 1:sqr3:2 triangle would be 3+sqr3= about 4.732

perimeter of a 1:1:sqr2 triangle would be 2+sqr2= about 3.414

ratio = (3+sqr3)/(2+sqr2) = about 4.732/3.414

UNLESS maybe you're comparing two right triangles of the same area

then keeping the 30-60-90 sides as 1-sqr3- 2 lengths

the 45-45-90 would become sqr(sqr3)-sqr(sqr3)- sqr2(sqr(sqr3))

perimeter of the 45-45-90 would then be (2+sqr2)sqr(sqr3)

= about 3.1414(sqr1.732)

= about 4.1343

ratio of perimeters = about 4.732/4.134