Question

ratios of similar right triangles
reset
scale factor n:
1
m∠a = 30° m∠c = 60°
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 sine - function definition

In a right - triangle, for an angle $\theta$, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$.

Step2: When scale factor $n = 1$

Given $\angle A=30^{\circ}$, by the sine - function definition, $\sin A=\sin30^{\circ}=\frac{1}{2}$. So the ratio of the side length of the side opposite $\angle A$ and the length of the hypotenuse is $\frac{1}{2}$.

Step3: When scale factor $n = 3$

The angles of similar right - triangles are the same. Since $\angle A = 30^{\circ}$, and for similar right - triangles, the ratios of corresponding sides are equal. So $\sin A=\sin30^{\circ}=\frac{1}{2}$. The ratio of the side length of the side opposite $\angle A$ to the length of the hypotenuse is still $\frac{1}{2}$.

Step4: General case for $30^{\circ}$ angle

For any right - triangle with a $30^{\circ}$ angle, by the sine function $\sin30^{\circ}=\frac{1}{2}$. The ratio of the side length of the side opposite the $30^{\circ}$ angle and the length of the hypotenuse is $\frac{1}{2}$.

Answer:

When the scale factor is 1, the ratio is $\frac{1}{2}$. When the scale factor is 3, the ratio is $\frac{1}{2}$. The ratio of the side length of the side opposite any $30^{\circ}$ angle and the length of the hypotenuse is $\frac{1}{2}$.