Answer:
d) x=4√3; y=9
Step-by-step explanation:
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
