Problem:
The measures of the smallest angles of three different right triangles sum to 90∘. All three triangles have side lengths that are primitive Pythagorean triples. Two of them are 3-4-5 and 5-12-13. What is the perimeter of the third triangle?
Answer Choices:
A. 40
B. 126
C. 154
D. 176
E. 208
Solution:
Let the smallest angle of the 3-4-5 triangle have measure α, the smallest angle of the 5−12−13 triangle have measure β, and the smallest angle of the third triangle have measure γ. It is given that α+β+γ=90∘, so cos(α+β+γ)=0. Expanding gives
cos(α+β+γ)=cos(α+β)cosγ−sin(α+β)sinγ=(cosαcosβ−sinαsinβ)cosγ−(sinαcosβ+cosαsinβ)sinγ=cosαcosβcosγ−sinαsinβcosγ−sinαcosβsinγ−cosαsinβsinγ=0
Because sinα=53,cosα=54,sinβ=135, and cosβ=1312,
cos(α+β+γ)=54⋅1312cosγ−53⋅135cosγ−53⋅1312sinγ−54⋅135sinγ=6548cosγ−6515cosγ−6536sinγ−6520sinγ=6533cosγ−6556sinγ=0.
Therefore 33cosγ=56sinγ, so tanγ=5633. Because 33 and 56 are relatively prime, perhaps they are the lengths of the two legs of the third triangle. Indeed, 332+562=1089+3136=4225=652, and the perimeter of the triangle is 33+56+65=(C)154.
OR
Let α,β, and γ be as in the first solution. Using complex numbers in polar form, note that 4+3i= 5cisα. Similarly, 12+5i=13 cis β. Multiplying these two quantities yields
(4+3i)(12+5i)=33+56i=65cis(2π−γ),
where the triangle with sides of length 33,56 , and 65 has angle γ opposite the side of length 33 and angle 2π−γ opposite the side of length 56 . Notice that
(5cisα)⋅(13cisβ)⋅(65cisγ)=(4+3i)(12+5i)(56+33i)=652i=652cis(α+β+γ)=652cis2π
The third triangle has sides of length 33,56 , and 65 , and 33+56+65=(C)154.
Note: A fun fact about the 33−56−65 triangle is that it is the smallest primitive Pythagorean triple that contains no prime values.
The problems and solutions on this page are the property of the MAA's American Mathematics Competitions