- If a medium side of a 30-60-90 triangle is 2sqrt3 in length, find the lengths of the remaining sides.

let the sides of the triangle be a, b and c and a<b<c, a and b are the legs, c is the hypotenuse

since opposite bigger angle lies bigger side follows the medium side b lies opposite 60 degrees angle

b = 2*sqrt3

sin (60) = b/c

c = b/sin(60)

sin 60 =(1/2)sqrt3

c = 2*sqrt3/((1/2)sqrt3)

c = 4

sin (30) = b/c

b = c sin(30)

b = 4*(1/2)

b = 2

the lengths of the remaining sides are 2 and 4.

- If the longest side in a 45-45-90 triangle is 12, find the remaining sides.

let the sides of the triangle be a, b and c and a<b<c, a>0, a and b are the legs, c is the hypotenuse

since opposite bigger angle lies bigger side follows c = 12

the triangle is isosceles this a = b

using the Pythagorean Theorem we have

a^2 + b^2 = c^2

2a^2 = 12^2

by solving we find

a = 8.49

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the remaining sides have lengths of 8.49 each.