45-45-90 Special Right Triangle Worksheet Answers

Mastering the 45-45-90 Special Right Triangle: A Comprehensive Guide with Worksheet Answers

Understanding special right triangles is crucial for success in geometry and trigonometry. Among these special triangles, the 45-45-90 triangle, also known as an isosceles right triangle, holds a significant place due to its predictable ratios and simplified calculations. This article provides a comprehensive guide to solving problems involving 45-45-90 triangles, including a detailed explanation of its properties, step-by-step problem-solving methods, and answers to a practice worksheet. This guide aims to build a strong foundational understanding of this essential geometric concept.

Understanding the 45-45-90 Triangle

A 45-45-90 triangle is a right-angled triangle where two of its angles measure 45 degrees each, and the third angle, naturally, measures 90 degrees. The key characteristic that makes this triangle "special" is its consistent side ratios. Because of its isosceles nature (two equal sides), the legs (the sides adjacent to the right angle) are congruent. This inherent symmetry simplifies calculations significantly.

Key Properties:

• Angles: 45°, 45°, 90°
• Sides: The two legs are congruent (equal in length). The hypotenuse (the side opposite the right angle) is √2 times the length of each leg.

This ratio of 1:1:√2 is the cornerstone of solving problems involving 45-45-90 triangles. Understanding this ratio allows for quick and efficient determination of unknown side lengths, given one known side.

Solving 45-45-90 Triangle Problems: A Step-by-Step Approach

The following steps outline a systematic approach to solving problems related to 45-45-90 triangles:

Step 1: Identify the Known Information

Begin by carefully examining the problem and identifying the given information. This might include the length of one side (leg or hypotenuse) or the relationship between sides.

Step 2: Apply the 1:1:√2 Ratio

Remember the fundamental ratio of a 45-45-90 triangle: leg : leg : hypotenuse = 1 : 1 : √2. Use this ratio to set up a proportion.

Step 3: Set up a Proportion

Based on the known information and the 1:1:√2 ratio, create a proportion to solve for the unknown side length(s).

Step 4: Solve the Proportion

Solve the proportion using algebraic methods. This often involves cross-multiplication or other simplification techniques.

Step 5: Check Your Answer

Always check your answer to ensure it is reasonable and consistent with the given information and the properties of a 45-45-90 triangle. Does the calculated side length make sense in relation to the other sides?

Example Problems and Solutions

Let's illustrate the process with a few examples:

Example 1:

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

Solution:

• Step 1: Known information: Leg = 5 cm
• Step 2: Ratio: leg : hypotenuse = 1 : √2
• Step 3: Proportion: 5/x = 1/√2
• Step 4: Solving for x (hypotenuse): x = 5√2 cm

Example 2:

A 45-45-90 triangle has a hypotenuse of length 10 inches. Find the length of each leg.

Solution:

• Step 1: Known information: Hypotenuse = 10 inches
• Step 2: Ratio: leg : hypotenuse = 1 : √2
• Step 3: Proportion: x/10 = 1/√2
• Step 4: Solving for x (leg): x = 10/√2 = 5√2 inches

Example 3:

In a 45-45-90 triangle, the area is 32 square meters. Find the length of each leg.

Solution:

• Step 1: Known information: Area = 32 sq m
• Step 2: Area of a triangle = (1/2) * base * height. In a 45-45-90 triangle, base = height = leg length (x).
• Step 3: Equation: (1/2) * x * x = 32
• Step 4: Solving for x: x² = 64, x = 8 meters (leg length)

45-45-90 Triangle Worksheet and Answers

Now, let's put your knowledge to the test with a worksheet. Remember to use the steps outlined above to solve each problem.

Worksheet:

1. A 45-45-90 triangle has a leg of length 7 cm. Find the length of the hypotenuse.
2. A 45-45-90 triangle has a hypotenuse of length 14 cm. Find the length of each leg.
3. The legs of a 45-45-90 triangle are each 12 inches long. Find the length of the hypotenuse.
4. The hypotenuse of a 45-45-90 triangle is 8√2 meters. Find the length of each leg.
5. A square has a diagonal of length 10√2 feet. What is the length of each side of the square? (Hint: A diagonal of a square divides it into two 45-45-90 triangles.)
6. Find the area of a 45-45-90 triangle with a leg of length 6 inches.
7. If the area of a 45-45-90 triangle is 50 square centimeters, find the length of its hypotenuse.
8. A 45-45-90 triangle has a hypotenuse of 20 cm. Calculate the perimeter.
9. Two legs of a 45-45-90 triangle measure x cm each. Express the perimeter in terms of x.
10. The perimeter of a 45-45-90 triangle is 12 + 12√2 cm. What is the length of its hypotenuse?

Answers:

1. 7√2 cm
2. 7√2 cm
3. 12√2 inches
4. 8 meters
5. 10 feet
6. 18 square inches
7. 10√2 centimeters
8. 20 + 20√2 cm
9. 2x + x√2 cm
10. 12 cm

Further Exploration and Applications

The 45-45-90 triangle is more than just a geometric concept; it has practical applications across various fields. Understanding its properties is crucial in:

• Construction and Engineering: Calculating distances, angles, and structural stability.
• Navigation and Surveying: Determining distances and bearings.
• Computer Graphics and Game Design: Modeling and rendering three-dimensional objects.
• Physics and Engineering: Solving problems related to forces, vectors, and motion.

This versatile triangle serves as a building block for more complex geometric problems and is instrumental in numerous real-world applications.

Conclusion

Mastering the 45-45-90 special right triangle is a fundamental skill for anyone studying geometry and trigonometry. By understanding its inherent properties and applying the 1:1:√2 ratio, you can efficiently solve a wide variety of problems. The provided worksheet and answers offer valuable practice, allowing you to solidify your understanding and build confidence in tackling more complex geometric challenges. Remember to practice regularly and apply these concepts to various real-world scenarios to enhance your understanding and problem-solving abilities.