Triangle similarity help?

Angle Bisector creates angles of θ and θ at the vertex and Φ and π-Φ on the opposing sides corresponding with the lengths 4 and 6 respectively.

The perimeter is 30 m, so let the sides opposite Φ and π-Φ be 10-x and 10+x respectively.

By the Sine Rule: 4/sinθ = (10-x)/sinΦ and 6/sinθ = (10+x)/sin(π-Φ) = (10+x)/sinΦ {as sin(π-Φ) = sinΦ}

So sinθ/sinΦ = 4/(10-x) = 6/(10+x), so (10-x):(10+x) = 4:6, so x=2,

and the sides are 8m and 12m respectively

Image: https://www.desmos.com/calculator/sk7wsvakxq

The Angle-Bisector theorem states that if a ray bisects an angle of a triangle, then it divides the opposite side into segments that are proportional to the other two sides.

Let the two unknown sides be a and b.

a/b = 4/6

a/b = 2/3

And all the sides add up to 30:

a + b + 4 + 6 = 30

a + b = 20

Solve the first equation in terms of a:

a = (2/3)b

Substitute into the second equation and solve for b:

(2/3)b + b = 20

(5/3)b = 20

b = 20 * 3/5

b = 12

Now solve for a:

a + b = 20

a + 12 = 20

a = 8

Answer:

8 m (on the side nearest 4 m)

12 m (on the side nearest 6 m)