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° (triangle with b right angle, bc=1, ab=√3, ac=2, scale factor: n = 1 dropdown)

Explanation:

First Question (Scale Factor = 1)

Step1: Identify sides

In the right triangle, \(\angle A = 30^\circ\), side opposite \(\angle A\) is \(BC = 1\), hypotenuse \(AC = 2\).

Step2: Calculate ratio

Ratio = \(\frac{\text{opposite to } \angle A}{\text{hypotenuse}} = \frac{1}{2}\).

Second Question (Scale Factor = 3)

Step1: Scale the sides

When scale factor is 3, new opposite side = \(1\times3 = 3\), new hypotenuse = \(2\times3 = 6\).

Step2: Calculate ratio

Ratio = \(\frac{3}{6} = \frac{1}{2}\).

Third Question (Any \(30^\circ\) angle)

Step1: Recall properties

In a 30 - 60 - 90 triangle, the side opposite \(30^\circ\) is half the hypotenuse (from triangle properties).

Step2: Determine ratio

So ratio is \(\frac{1}{2}\).

Answer:

s:

• When scale factor is 1: \(\frac{1}{2}\)
• When scale factor is 3: \(\frac{1}{2}\)
• For any \(30^\circ\) angle: \(\frac{1}{2}\)