In a 30 ^(∘) - 60 ^(∘) - 90 ^(∘) triangle, the hypotenuse h is 2 times the length of the shorter leg s, and the longer leg l is sqrt(3) times the length of the shorter leg.
Let's name the vertices of the given figure. Then we can find x,y and z, one at a time.
Finding x
We are given that ∠ CAD has a measure of 45^(∘). Therefore, △ ACD is a 45^(∘)-45 ^(∘)-90 ^(∘) triangle.
The length of the leg is x and the length of the hypotenuse is 18. By the 45^(∘)-45 ^(∘)-90 ^(∘) Triangle Theorem, the length of the hypotenuse is sqrt(2) times the length of a leg, x.
x sqrt(2) = 18
Let's solve the above equation for x.
We are given that ∠ ABC has a measure of 60^(∘). Therefore, △ ABC is a 30^(∘)-60^(∘)-90 ^(∘) triangle.
The length of the shorter leg is y, and the length of the longer leg is 18. By the 30^(∘)-60 ^(∘)-90 ^(∘) Triangle Theorem, the length of the longer leg is sqrt(3) times the length of a shorter leg, y.
y sqrt(3) = 18
Let's solve the above equation for y.
To find the value of z, we will again use the 30^(∘)-60 ^(∘)-90 ^(∘) Triangle Theorem. According to this theorem, the length of the hypotenuse z is 2 times the length of the length of the shorter leg, 6sqrt(3).
z=2( 6sqrt(3))=12sqrt(3)
