Question

Draw and label a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle in which the sides are 5 inches, 10 inches, and \(5 \sqrt{3}\) inches.

Short answer

Draw a 30-60-90 triangle with a hypotenuse of 10 inches, a shorter side of 5 inches, and the last side as \(5 \sqrt{3}\) inches.

Step-by-step solution

Understanding the 30-60-90 Triangle

A 30-60-90 triangle is a special type of right triangle with angles measuring 30 degrees, 60 degrees, and 90 degrees. A key property of this triangle is its side ratios: the side opposite the 30-degree angle is half of the hypotenuse, and the side opposite the 60-degree angle is equal to the shorter side times the square root of 3.

Identify the Triangle Components

In the given triangle, the side lengths are 5 inches, 10 inches, and \(5 \sqrt{3}\) inches. The hypotenuse is the longest side, which in this case is 10 inches. The side opposite the 30-degree angle is 5 inches (half of 10), and the side opposite the 60-degree angle is \(5 \sqrt{3}\) inches.

Draw the Triangle

Begin by drawing the hypotenuse which is 10 inches. Then, from one end of the hypotenuse, draw a line measuring 5 inches perpendicular to the baseline to represent the side opposite to the 30-degree angle. Connect the endpoint of this 5-inch line to the other end of the hypotenuse to form the side opposite the 60-degree angle, which should measure \(5 \sqrt{3}\) inches.

Label the Triangle

Label the hypotenuse as 10 inches. The side opposite the 30-degree angle is labeled as 5 inches, and the side opposite the 60-degree angle is labeled as \(5 \sqrt{3}\) inches. Mark the 30-degree angle opposite the 5-inch side and the 60-degree angle opposite the \(5 \sqrt{3}\) inch side.

Verify the Triangle

Check that the triangle's properties align with known 30-60-90 triangle properties. Ensure the side opposite the 30-degree angle is half the hypotenuse (5 inches is half of 10 inches), and the side opposite the 60-degree angle matches \(5 \sqrt{3}\), which is the shorter side multiplied by \(\sqrt{3}\).

Key concepts

Right Triangle Properties

A right triangle is defined by having one angle equal to 90 degrees. This creates two smaller angles that add up to 90 degrees. The longest side in a right triangle, opposite the right angle, is known as the hypotenuse. The two other sides are called the legs of the triangle. Right triangles are crucial in geometry as they help us understand relationships between angles and side lengths.

• The Pythagorean Theorem applies to all right triangles, allowing us to calculate the length of one side when the lengths of the other two are known: \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse.
• In a 30-60-90 triangle, which is a specific type of right triangle, unique properties govern the side lengths due to the fixed angles.

Right triangles often appear in real-world problems and are fundamental in trigonometry.

Triangle Side Ratios

In geometry, the concept of triangle side ratios helps describe how the lengths of the sides of a triangle relate to each other. This is particularly useful in right triangles, such as the 30-60-90 triangle.

• For a 30-60-90 triangle, an important ratio exists: the sides are in a 1: \(\sqrt{3}\) : 2 relationship.
• This means that the shorter leg (opposite the 30-degree angle) is half the length of the hypotenuse. The longer leg (opposite the 60-degree angle) is the shorter leg times \( \sqrt{3} \).

Understanding these ratios allows you to determine the lengths of all sides in the triangle, given just one side. By mastering side ratios, you gain a deeper comprehension of how right triangles function and can efficiently solve related problems.

Special Right Triangles

Special right triangles, like the 30-60-90 and the 45-45-90 triangles, have set angle measures and predictable side ratios. This makes them powerful tools in quickly solving geometry problems.

• The 30-60-90 triangle has angles of 30, 60, and 90 degrees. Its side lengths follow a specific ratio, making it unique and simple to work with.
• These triangles often appear in problems where simplifying calculations is required, since their properties reduce complexity.

By learning about these special triangles, such as their fixed angle measures and corresponding ratios, you enhance your ability to handle various geometric tasks easily.

Geometry Concepts

Geometry, the branch of mathematics dealing with shapes, sizes, and properties of figures, is essential for understanding the world around us. One key aspect is the study of triangles, especially right triangles.

• In a typical geometry problem involving triangles, you'll use properties such as angle sums, congruency, and similarity to solve or prove aspects about the figure.
• Understanding the characteristics of special triangles can greatly simplify certain computations and proofs.

Geometry is not just about calculations; it's also about visualizing and proving concepts. By studying geometry, students gain critical thinking skills that apply to real-world problems, enhancing logical reasoning wherever they go.