Question

rizing 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 any 30 - angle and the length of the hypotenuse?

Explanation:

Step1: Recall trigonometric ratio definition

In a right - triangle, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse, i.e., $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$.

Step2: Identify the angle

We are given $\angle A = 30^{\circ}$.

Step3: Use the sine value of 30 - degree angle

The sine of a $30^{\circ}$ angle is a well - known trigonometric value. $\sin30^{\circ}=\frac{1}{2}$. This ratio is constant for any right - triangle with a $30^{\circ}$ angle, regardless of the scale factor. When the scale factor is 1 or 3 or any non - zero value, the ratio of the side length of the side opposite $\angle A$ (a $30^{\circ}$ angle) to the length of the hypotenuse is $\frac{1}{2}$.

Answer:

$\frac{1}{2}$