8-3 Practice Special Right Triangles Answer Key

8-3 Practice: Special Right Triangles – A Comprehensive Guide with Answers

Understanding special right triangles is crucial for success in geometry and trigonometry. These triangles, possessing unique angle measures of 30-60-90 and 45-45-90 degrees, exhibit consistent relationships between their sides. Mastering these relationships simplifies problem-solving and lays the groundwork for more advanced mathematical concepts. This guide provides a thorough walkthrough of 8-3 practice problems involving special right triangles, offering detailed solutions and explanations to solidify your understanding.

Understanding Special Right Triangles: The Foundation

Before diving into the practice problems, let's revisit the fundamental properties of 30-60-90 and 45-45-90 triangles. Remembering these ratios is key to quickly solving problems.

1. 45-45-90 Triangles (Isosceles Right Triangles):

• Angles: Two 45-degree angles and one 90-degree angle.
• Sides: The two legs are congruent (equal in length), and the hypotenuse is √2 times the length of a leg. This can be represented as: leg : leg : leg√2

2. 30-60-90 Triangles:

• Angles: Angles of 30, 60, and 90 degrees.
• Sides: The side opposite the 30-degree angle (shorter leg) is half the length of the hypotenuse. The side opposite the 60-degree angle (longer leg) is √3 times the length of the shorter leg. This relationship can be expressed as: x : x√3 : 2x, where x is the length of the shorter leg.

8-3 Practice Problems: Detailed Solutions

Let's tackle some example problems, applying the principles above. Remember to always draw a diagram to visualize the problem!

Problem 1: A 45-45-90 triangle has a leg of length 5 cm. Find the length of the hypotenuse.

Solution:

Since it's a 45-45-90 triangle, the ratio of sides is 1 : 1 : √2. We know one leg is 5 cm. Therefore:

• Leg = 5 cm
• Leg = 5 cm
• Hypotenuse = 5√2 cm

Therefore, the hypotenuse is 5√2 cm.

Problem 2: A 30-60-90 triangle has a hypotenuse of length 12 inches. Find the lengths of the other two sides.

Solution:

In a 30-60-90 triangle, the ratio is x : x√3 : 2x. The hypotenuse is 2x, which is given as 12 inches. So:

• 2x = 12 inches
• x = 6 inches (shorter leg)
• x√3 = 6√3 inches (longer leg)

Therefore, the shorter leg is 6 inches and the longer leg is 6√3 inches.

Problem 3: A right triangle has a hypotenuse of length 10 and one leg of length 5. Is this a special right triangle? If so, what type?

Solution:

We can use the Pythagorean theorem (a² + b² = c²) to check if this is a right triangle. If it is, we can then determine if it's a special right triangle.

• a² + b² = c²
• 5² + b² = 10²
• 25 + b² = 100
• b² = 75
• b = √75 = 5√3

The sides are 5, 5√3, and 10. This matches the ratio of a 30-60-90 triangle (x : x√3 : 2x, where x = 5).

Problem 4: Find the perimeter of a 45-45-90 triangle with a hypotenuse of 8 cm.

Solution:

First, find the length of each leg. The ratio is 1:1:√2. Let 'x' represent the length of a leg:

• x√2 = 8 cm
• x = 8/√2 = 4√2 cm

Both legs are 4√2 cm.

Perimeter = leg + leg + hypotenuse = 4√2 + 4√2 + 8 = 8 + 8√2 cm

Problem 5: A ramp has a slope of 30 degrees. If the ramp is 10 feet long, how high is the ramp's vertical rise?

Solution:

This problem describes a 30-60-90 triangle, where the ramp's length is the hypotenuse (10 feet), and the vertical rise is the side opposite the 30-degree angle.

Using the ratio x : x√3 : 2x, we have:

• 2x = 10 feet
• x = 5 feet

The vertical rise (opposite the 30-degree angle) is 5 feet.

Problem 6: Two legs of a right triangle are equal in length and measure 7 inches each. What is the length of the hypotenuse?

Solution:

This is a 45-45-90 triangle. Using the ratio 1:1:√2:

• Hypotenuse = leg * √2 = 7√2 inches

The hypotenuse measures 7√2 inches.

Problem 7: A square has a diagonal of length 12 cm. What is the length of each side?

Solution:

A diagonal of a square divides it into two 45-45-90 triangles. The diagonal is the hypotenuse. Using the ratio 1:1:√2:

• x√2 = 12 cm
• x = 12/√2 = 6√2 cm

Each side of the square measures 6√2 cm.

Problem 8: A triangle has sides of length 6, 6√3, and 12. Is this a special right triangle? If so, what type?

Solution: The ratio of the sides is 6 : 6√3 : 12, which simplifies to 1 : √3 : 2. This corresponds to the ratio of a 30-60-90 triangle.

Problem 9: Find the area of a 30-60-90 triangle with a hypotenuse of 8 cm.

Solution:

First, find the lengths of the legs. The ratio is x : x√3 : 2x.

• 2x = 8 cm
• x = 4 cm (shorter leg)
• x√3 = 4√3 cm (longer leg)

Area of a triangle = (1/2) * base * height = (1/2) * 4 cm * 4√3 cm = 8√3 cm²

Problem 10: A regular hexagon has a side length of 10 cm. Find the length of its apothem (the distance from the center to the midpoint of a side).

Solution: A regular hexagon can be divided into six equilateral triangles. The apothem is the altitude of one of these equilateral triangles. In an equilateral triangle with side length 10 cm, the apothem is:

Apothem = (10/2)√3 = 5√3 cm

The apothem of the hexagon is 5√3 cm.

Advanced Problem Solving Strategies

For more complex problems involving special right triangles, consider these advanced techniques:

• Combining Triangles: Often, a problem might involve multiple triangles, requiring you to solve for one triangle's components before moving to another. Break down the problem into manageable parts.
• Trigonometric Functions: While you can solve many special right triangle problems using the side ratios, understanding basic trigonometric functions (sine, cosine, tangent) can be helpful for more challenging scenarios.
• Algebraic Manipulation: Sometimes, you might need to use algebraic techniques to solve for unknown side lengths, such as solving simultaneous equations.

Practice Makes Perfect

The key to mastering special right triangles is consistent practice. Work through numerous problems, varying the given information and focusing on understanding the underlying principles. The more you practice, the faster and more accurately you'll be able to solve these types of problems. Remember to always draw diagrams and carefully label the sides and angles. With dedicated effort, you will develop a strong understanding of special right triangles and their applications.