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 y and x, one at a time.
Finding y
We are given that ∠ABC is a right angle and that the measure of ∠ACB is 45^(∘). Therefore, by the Triangle Angle Sum Theorem, the measure of ∠BAC must also be 45^(∘). This means, △ ABC is a 45^(∘) - 45 ^(∘) - 90 ^(∘) triangle. In such triangle, the hypotenuse is sqrt(2) times the length of a leg. Knowing that, we can find the value of y.
y= sqrt(2) * 6 ⇔ y=6sqrt(2)
Finding x
We are given that ∠BDC is a right angle, and the measure of ∠ACB is 45 ^(∘). Again, by the Triangle Angle Sum Theorem, the measure of ∠CBD must also be 45^(∘), which means △ BCD is a 45^(∘) - 45 ^(∘) - 90 ^(∘) triangle. Let's add the information to our diagram.
The length of the leg is x and the length of the hypotenuse is 6. By the 45^(∘) - 45 ^(∘) - 90 ^(∘) Triangle Theorem, the length of the hypotenuse is sqrt(2) times the length of a leg, x.
