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 to the length of the hypotenuse? m∠a = 30° m∠c = 60°

Explanation:

Step1: Identify side opposite ∠A

In the given triangle, $\angle A = 30^\circ$, the side opposite it is $BC = 1$, hypotenuse $AC = 2$.
Ratio: $\frac{BC}{AC} = \frac{1}{2}$

Step2: Scale factor = 3, recalculate sides

Scaled side opposite $\angle A$: $1 \times 3 = 3$
Scaled hypotenuse: $2 \times 3 = 6$
Ratio: $\frac{3}{6} = \frac{1}{2}$

Step3: Generalize for 30° angle

Similar triangles preserve side ratios, so the ratio is constant.

Answer:

1. When scale factor is 1: $\boldsymbol{\frac{1}{2}}$
2. When scale factor is 3: $\boldsymbol{\frac{1}{2}}$
3. For any 30° angle: $\boldsymbol{\frac{1}{2}}$