Question

1. identify the correct ratio for each side of the special right triangles

drag and drop the correct choice into the blank spaces above
a=
b=
c=
d=
e=
f=

Explanation:

Step1: Recall the properties of a 30 - 60 - 90 triangle

In a 30 - 60 - 90 triangle, the sides are in the ratio \(1:\sqrt{3}:2\). If the side opposite the 30° angle is \(x\), the side opposite the 60° angle is \(x\sqrt{3}\), and the hypotenuse is \(2x\).

Step2: Analyze the first triangle (general right - angled triangle) and relate to the 30 - 60 - 90 triangle

Let's assume in the 30 - 60 - 90 triangle, if we consider the side opposite 30° as \(a\) (in the general right - angled triangle, assume the side opposite the smaller non - right angle in the general triangle corresponds to the side opposite 30° in the 30 - 60 - 90 triangle).
The side opposite 60° in the 30 - 60 - 90 triangle (\(f\)) is \(\sqrt{3}\) times the side opposite 30° (\(a\)). The hypotenuse (\(d\)) is 2 times the side opposite 30° (\(a\)). The side adjacent to 30° (\(e\)) (which is opposite 60° in the angle sense of the 30 - 60 - 90 triangle properties) is \(\sqrt{3}\) times the side opposite 30°.

If we assume the side opposite the smaller non - right angle in the first triangle (let's say \(a\)) corresponds to the side opposite 30° in the 30 - 60 - 90 triangle:

• \(a=\text{opposite 30°}\), \(b = \text{opposite 60°}\), \(c=\text{hypotenuse}\)
• In the 30 - 60 - 90 triangle: \(a=\frac{1}{2}d\), \(b=\frac{\sqrt{3}}{2}d\), \(c = d\) (where we use the ratio \(a:b:c = 1:\sqrt{3}:2\) and \(d\) as the hypotenuse of the 30 - 60 - 90 triangle).

Answer:

\(a=\frac{1}{2}d\), \(b=\frac{\sqrt{3}}{2}d\), \(c = d\), \(e=\sqrt{3}a\), \(f=a\)