Question

Find the maximum perimeter of a right triangle with hypotenuse 12cm

Answer 1

If the hypotenuse is 12, then lengths of the legs are: 
a = 12sinΘ 
b = 12cosΘ 
The perimeter of the triangle is: 
P = 12 + 12(sinΘ + cosΘ) 
dP/dΘ = 12(cosΘ - sinΘ) = 0 
cosΘ = sinΘ 
tanΘ = 1 ==> Θ=45° 
the sum of the angles of a triangle is always 180. the third unknown angle is 180 - 90 - 45 = 45. 
The three angles of the triangle which maximize the perimeter of a right angle for a given hypotenuse are, 90°, 45°, and 45°. 

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Answer 2

HEY MATE HERE IS YOUR ANSWER--

If the hypotenuse is 12, then lengths of the legs are:

a = 12sinΘ

b = 12cosΘ

The perimeter of the triangle is:

P = 12 + 12(sinΘ + cosΘ)

dP/dΘ = 12(cosΘ - sinΘ) = 0

cosΘ = sinΘ

tanΘ = 1 ==> Θ=45°

the sum of the angles of a triangle is always 180. the third unknown angle is 180 - 90 - 45 = 45.

The three angles of the triangle which maximize the perimeter of a right angle for a given hypotenuse are, 90°, 45°, and 45°.