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 ∠ ADC is a right angle, and sides AB and AD are equal. Therefore, the polygon ABCD is a square, which means all of the sides are equal. Let's add this information to our diagram.
Because polygon ABCD is a square, we know that ∠ BAD is a right angle and that sides AB and AD are congruent. Therefore, △ ABD is a 45^(∘) - 45 ^(∘) - 90 ^(∘) triangle. In such triangle, the hypotenuse is sqrt(2) times the length of a leg. With this information, we can find the value of y.
y= sqrt(2) * 12 ⇔ y=12sqrt(2)
Finding x
Let's add the obtained information to the diagram.
Because polygon ABCD is a square, we know that ∠ ABC is a right angle. Therefore, ∠ ABD and ∠ CBD are complementary. Knowing the measure of ∠ ABD equals 45^(∘), we can find the value of x.
90=45+x ⇔ x=45
