Question

Which of the following sets of 3 numbers could be the side lengths, in meters, of a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle? A. \(1,1,1\) B. \(1,1, \sqrt{2}\) C. \(1, \sqrt{2}, \sqrt{2}\) D. \(1, \sqrt{2}, \sqrt{3}\) E. \(1, \sqrt{3}, 2\)

Short answer

Answer: Side lengths 1, √3, and 2.

Step-by-step solution

Check Option A

Option A provides side lengths of 1, 1, and 1. Since these lengths are all equal, this set of side lengths cannot form a 30-60-90 triangle, as it would instead form an equilateral triangle with all angles equal to \(60^{\circ}\).

Check Option B

Option B provides side lengths of 1, 1, and \(\sqrt{2}\). Compare these values to the ratio 1:\(\sqrt{3}\):2. These side lengths do not meet this ratio requirement, as the middle side length is not equal to \(\sqrt{3}\). Therefore, this set of side lengths cannot form a 30-60-90 triangle.

Check Option C

Option C provides side lengths of 1, \(\sqrt{2}\), and \(\sqrt{2}\). Compare these values to the ratio 1:\(\sqrt{3}\):2. These side lengths do not meet this ratio requirement, as the last side length is not equal to 2. Therefore, this set of side lengths cannot form a 30-60-90 triangle.

Check Option D

Option D provides side lengths of 1, \(\sqrt{2}\), and \(\sqrt{3}\). Compare these values to the ratio 1:\(\sqrt{3}\):2. These side lengths do not meet this ratio requirement, as the middle side length is not equal to \(\sqrt{3}\) and the last side length is not equal to 2. Therefore, this set of side lengths cannot form a 30-60-90 triangle.

Check Option E

Option E provides side lengths of 1, \(\sqrt{3}\), and 2. Compare these values to the ratio 1:\(\sqrt{3}\):2. These side lengths meet the 1:\(\sqrt{3}\):2 ratio requirement, and therefore, this set of side lengths can form a 30-60-90 triangle. In conclusion, the correct answer is option E - the side lengths 1, \(\sqrt{3}\), and 2 can form a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle.

Key concepts

Geometry of the 30-60-90 Triangle

Understanding the geometry of the 30-60-90 triangle is essential to grasping how special right triangles work. A 30-60-90 triangle is a right triangle characterized by interior angles measuring 30 degrees, 60 degrees, and the remaining 90 degrees. This specific angle combination gives the triangle its unique properties and predictable side lengths.

When working with a 30-60-90 triangle, the side opposite the 30-degree angle is always the shortest, often referred to as the 'short leg'. We denote this side as 'a'. The side opposite the 60-degree angle is \(\sqrt{3}\) times longer than 'a' and is called the 'long leg', denoted as 'b'. Finally, the hypotenuse, which is opposite the 90-degree angle, is exactly twice the length of the short leg, and denoted as 'c'. This gives us the ratio of the sides: \(a : b : c = 1 : \sqrt{3} : 2\).

These exact proportions simplify calculations and problem solving involving such triangles. It bears repeating: when dealing with 30-60-90 triangles, you only need to know the length of one side to determine the lengths of the other two sides, thanks to this constant ratio.

Trigonometry and the 30-60-90 Triangle

Trigonometry, often dubbed as the study of triangles, dives deep into the relationships between the sides and angles of triangles. The 30-60-90 triangle is particularly important in trigonometry because the sine, cosine, and tangent of these specific angles lead to simple, rational numbers. This makes calculations more manageable and understandable.

In a 30-60-90 triangle, the sine of the 30-degree angle is \(\frac{1}{2}\), because sine is the ratio of the opposite side over the hypotenuse ( \(\frac{a}{c}\)). Meanwhile, the cosine of the 30-degree angle is \(\frac{\sqrt{3}}{2}\), since cosine is the adjacent side over the hypotenuse ( \(\frac{b}{c}\)).

These trigonometric values are particularly handy when it comes to solving problems without the actual measurements of the sides. Understanding these values within the context of the special 30-60-90 triangle's side ratios enriches one's ability to make quick and accurate computations in geometry and beyond.

Special Right Triangles

The category of special right triangles includes the 30-60-90 triangle as well as the 45-45-90 triangle. These triangles are regarded 'special' because of their predictable angles and side ratio relationships. Memorizing these ratios allows students to quickly solve a variety of geometry problems without having to go through complex calculations every time.

For the 30-60-90 triangle, we have already established the side length ratio of \(1 : \sqrt{3} : 2\). It's crucial to note that these ratios hold true no matter the size of the triangle; it could be minuscule or massive, and yet, the relationships between the side lengths will always be constant.

This property of special right triangles is what makes them indispensable in more advanced mathematics, including proofs, trigonometric identities, and even in practical applications like construction and architecture. By understanding and applying the concepts of special right triangles, students can solve complex geometric problems with efficiency and ease.