Question

ratios of similar right triangles
create similar right triangles by changing the scale factor of the right triangle.
when the scale factor is 1, what is the ratio of the side length of the side opposite ∠a and the length of the hypotenuse?
change the scale factor to 3. what is the ratio of the side length of the side opposite ∠a to the length of the hypotenuse?
what is the ratio of the side length of the side opposite and the length of the hypotenuse?

Explanation:

Step1: Recall sine - function property

For a right - triangle, $\sin A=\frac{\text{opposite}}{\text{hypotenuse}}$. For $\angle A = 30^{\circ}$, $\sin30^{\circ}=\frac{1}{2}$.

Step2: Understand similarity of right - triangles

Similar right - triangles have the same angle measures. The ratio of the side opposite $\angle A$ to the hypotenuse is given by the sine of $\angle A$. Since the angle measure of $\angle A$ remains $30^{\circ}$ even when the scale factor changes, the ratio $\frac{\text{opposite}}{\text{hypotenuse}}$ remains the same.

Answer:

$\frac{1}{2}$