Triangle RSU is an equilateral triangle. RT bisects US. Find x and y.

Using the fact that the altitude of an isosceles triangle (or equilateral triangle) bisects the top angle and base we know tht RT is an altitude, therefore:

RT⊥TS (definition of altitude), UT=ST=4 (definition of bisector)

And since UT+ST = US, US=4+4=8

And since this is an equilateral triangle RS=8

Therefore to find x we can use right triangle RTS and use the pythagorean theorem to find:

RT²+ST²=RS²

Substitute:

⇒ x²+4²=8²

Arithmetic:

⇒x²=48

Square root both sides:

⇒x=√48 = √16*3=4√3 (square root simplification)

Now to find y using the fact that RT⊥ST:

m∠RTS=90° (definition of perpendicular)

m∠TRS=(2y+12)° (given)

m∠RST=2·m∠TRS=2(2y+12)°=(4y+24)°

Because of the fact that the measures of the angles in a triangle add to 180° in triangle RST:

m∠RTS+m∠RST+m∠TRS=180°

Substitution:

⇒ 90°+2y+12+4y+24=180°

Add like terms:

⇒ 6y+126=180

Subtract 126:

⇒ 6y=54°

Divide by 6:

⇒ y=9

A faster method for y:

Since the angles of an equilateral triangle each are 60°:

2y+12 = 60/2 (definition of bisector)

⇒2y+12=30

Subtract 12:

⇒ 2y=18

Divide by 2:

y=9