Question

8.6 special right triangles 30-60-90
7)
find the values of x and y for the right triangle with one angle 30°, one leg length 6, hypotenuse x, and the other leg y.
$x=\square\sqrt{\square}$
$y=\square\sqrt{\square}$

Explanation:

Step1: Identify side roles in 30-60-90 triangle

The side adjacent to 30° is 6, so it is the longer leg. In a 30-60-90 triangle: longer leg = $\text{shorter leg} \times \sqrt{3}$, hypotenuse = $2 \times \text{shorter leg}$.

Step2: Solve for shorter leg $y$

Rearrange longer leg formula: $y = \frac{\text{longer leg}}{\sqrt{3}} = \frac{6}{\sqrt{3}} = 2\sqrt{3}$

Step3: Solve for hypotenuse $x$

Use hypotenuse formula: $x = 2y = 2 \times 2\sqrt{3} = 4\sqrt{3}$

Answer:

$x=4\sqrt{3}$
$y=2\sqrt{3}$