which of the following are true statements about a 30-60-90 triangle?
The correct answer and explanation is:
A 30-60-90 triangle is a right triangle with angles measuring 30°, 60°, and 90°. This type of triangle has a distinct set of properties that make it unique.
Correct Answer:
In a 30-60-90 triangle, the relationships between the lengths of the sides are consistent and based on a fixed ratio. If the shortest side, opposite the 30° angle, has a length of x, then the length of the side opposite the 60° angle is x√3, and the length of the hypotenuse (opposite the 90° angle) is 2x.
Explanation:
The key to understanding a 30-60-90 triangle is recognizing the proportional relationship between the sides. In such triangles, there are fixed ratios that apply regardless of the size of the triangle. These ratios are derived from the properties of a right triangle and the specific angles involved.
- Short Side (opposite 30°): The side opposite the 30° angle is always the shortest side. Let’s call it x. This is the baseline for determining the lengths of the other two sides.
- Longer Leg (opposite 60°): The side opposite the 60° angle is always longer than the shorter leg. It is x√3 in length. This relationship comes from applying trigonometric ratios like sine and cosine to the angles.
- Hypotenuse (opposite 90°): The longest side of the triangle, opposite the 90° angle, is always twice the length of the shortest side. Therefore, the hypotenuse is 2x.
For example, if the shortest side is 3 units, then:
- The longer leg would be 3√3 ≈ 5.196 units.
- The hypotenuse would be 2 * 3 = 6 units.
These proportions hold true for any 30-60-90 triangle, and they make these triangles particularly easy to solve when you know the length of just one side. The geometric properties and the predictable relationships between the sides make the 30-60-90 triangle an important shape in trigonometry and geometry.