Question

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? m∠a = 30°. m∠c = 60°. scale factor: n = 1

Explanation:

Step1: Recall sine - ratio property

In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. For $\theta = 30^{\circ}$, $\sin30^{\circ}=\frac{1}{2}$, so the ratio of the side opposite a $30^{\circ}$ angle to the hypotenuse is $\frac{1}{2}$.

Step2: Consider similar - triangle property

When the scale factor changes, the ratios of corresponding sides in similar triangles remain the same. So when the scale factor is 3, the ratio of the side opposite $\angle A$ to the hypotenuse is still $\frac{1}{2}$. When the scale factor is 1, the ratio of the side opposite $\angle A$ to the hypotenuse is $\frac{1}{2}$.

Answer:

$\frac{1}{2}$; $\frac{1}{2}$; $\frac{1}{2}$