Question

Hello!


If the angles of a triangle are in the ratio of 3:4:5 , then ratio of its sides is

Answer 1

Hey there!

We know that, Sum of angle in a triangle= 180 °

3x + 4x + 5x = 180

12x = 180

x = 15°

Angles are 45 , 60 , 75 .

According to Law of sines :

a/sinA = b/sinB = c/sinC .

a/sinA = b/sinB.

a/(1/√2 ) = b/ (√3/2)

√2a = 2b/√3

a/b = √2 : √3

a/b = 2 : √6

____________________________________________

b/sin60 = c/sin75

2b/√3 = 2√2c /( √3 + 1 )

b/√3 = √2c / √3 + 1

b /c = √6 / √3 + 1

Now, a : b : c = 2 : √6 : √3 + 1 .

Hope helped!

Answer 2

 Part 1
For any triangle with sides a,b,c and corresponding opposite angles A,B,C the Law of Sines tells us that

sin(A)a=sin(B)b=sin(C)c

or, by rearranging:

ba=sin(B)sin(A)

and

céa=sin(C)sin(A)

Part 2
If ∠A:∠B:∠C are in the ratio 3:4:5
since ∠A+∠B+∠C=π

∠A=3π12=π4

∠B=4π12=π3

∠C=5π12

Part 3
a:b:c
=aa:ba:ca

=1:sin(B)sin(A):sin(C)sin(A)

=1:sin(π3)sin(π4):sin(5π12)sin(π4)

(and using my calculator):
=1:1.224745:1.366025