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 trigonometric ratio for 30 - degree angle

In a right - triangle with an angle of 30 degrees, the ratio of the side opposite the 30 - degree angle to the hypotenuse is given by the sine function. $\sin(A)=\frac{\text{opposite}}{\text{hypotenuse}}$. When $A = 30^{\circ}$, $\sin(30^{\circ})=\frac{1}{2}$. So the ratio of the side opposite $\angle A$ to the hypotenuse is $\frac{1}{2}$.

Step2: Consider the effect of scale factor

For similar triangles, the ratios of corresponding sides are equal. When the scale factor is 1, the ratio of the side opposite $\angle A$ to the hypotenuse is $\frac{1}{2}$. When the scale factor is changed to 3, the ratio of the side opposite $\angle A$ to the hypotenuse remains $\frac{1}{2}$ because the triangles are similar and the trigonometric ratios depend only on the angles.

Answer:

When the scale factor is 1, the ratio of the side opposite $\angle A$ to the hypotenuse is $\frac{1}{2}$. When the scale factor is 3, the ratio of the side opposite $\angle A$ to the hypotenuse is $\frac{1}{2}$.