In Lesson 13.3, we will explore special right triangles and provide an answer key to help you with your understanding of the topic. Special right triangles are a type of right triangle where the angles and side lengths have specific relationships that make them easier to solve for. These relationships can be used to find missing side lengths or angles.
There are two types of special right triangles: 45-45-90 and 30-60-90 triangles. In a 45-45-90 triangle, the two legs are congruent and the hypotenuse is equal to the length of a leg multiplied by the square root of 2. In a 30-60-90 triangle, the shorter leg is half the length of the hypotenuse, and the longer leg is the length of the shorter leg multiplied by the square root of 3.
Using these relationships, you can solve for missing side lengths or angles in special right triangles. By applying the Pythagorean theorem and the ratios of the sides, you can easily find the solutions to various problems involving special right triangles. The answer key provided will guide you through the steps to solve these problems and help you check your answers.
By understanding the properties of special right triangles and using the answer key as a guide, you will be able to confidently solve problems involving these triangles. This will not only help you in geometry but also in other areas of mathematics where special right triangles are applicable, such as trigonometry. So, let’s dive into Lesson 13.3 and explore the world of special right triangles!
Lesson 13.3 Special Right Triangles Answer Key
After completing Lesson 13.3 on special right triangles, it’s time to check your understanding with the answer key. This answer key will provide you with the correct solutions for the problems in the lesson, helping you evaluate your performance and identify any areas that need further review.
Using the answer key, you can compare your work with the correct answers and see if you arrived at the same conclusions. If your answers match, it’s a sign that you have a strong understanding of the material. If your answers differ, take the time to review the steps and concepts covered in the lesson. Pay attention to where you might have made a mistake and try to understand why.
To help you further, the answer key may also include explanations for the correct solutions. These explanations can provide insights into the thought process behind solving each problem, helping you develop a deeper understanding of the concepts. Make sure to read and understand these explanations, even if your answer is correct, as they can offer valuable insights.
Remember, the answer key is a tool to assist you in your learning. It’s important to not just rely on the answer key alone, but to actively engage with the material, practice problem-solving, and seek guidance from your instructor or classmates if needed. Through consistent practice and reinforcement, you will be able to master the concepts of special right triangles and confidently apply them to solve various mathematical problems.
Here’s a sample answer key for a lesson on special right triangles:
- Problem 1: Sin∠C = √3/2
- Problem 2: 30°
- Problem 3: 2√3
- Problem 4: 45°
- Problem 5: 6√2
Remember to check the complete answer key provided by your instructor to get the correct answers for all the problems in Lesson 13.3. Good luck with your studies!
Overview
The lesson on special right triangles focuses on two types of triangles: the 45-45-90 triangle and the 30-60-90 triangle. These triangles have special properties that allow us to easily find the lengths of their sides and the measures of their angles.
45-45-90 Triangle: This type of triangle is an isosceles right triangle, meaning it has two congruent sides and one right angle. The two congruent sides have a 45-degree angle opposite them. The hypotenuse, which is the side opposite the right angle, can be found by multiplying the length of one of the congruent sides by the square root of 2.
30-60-90 Triangle: This type of triangle is a right triangle with angles measuring 30 degrees, 60 degrees, and 90 degrees. The side opposite the 30-degree angle is half the length of the hypotenuse, and the side opposite the 60-degree angle is the length of the hypotenuse multiplied by the square root of 3.
By understanding the ratios and properties of these special right triangles, we can quickly solve problems involving these triangle types and use them as shortcuts in geometry and trigonometry.
Some key concepts to remember are:
- The sides of a 45-45-90 triangle are in a ratio of 1:1:√2.
- The sides of a 30-60-90 triangle are in a ratio of 1:√3:2.
- The angles of a 45-45-90 triangle are 45 degrees, 45 degrees, and 90 degrees.
- The angles of a 30-60-90 triangle are 30 degrees, 60 degrees, and 90 degrees.
Understanding and being able to recognize these special right triangles can greatly simplify problem-solving in geometry and trigonometry, saving both time and effort.
Definition of Special Right Triangles
A special right triangle is a triangle that has some unique characteristics and properties. In particular, special right triangles are defined by their angles and side lengths, making them distinct from other types of triangles.
There are two main types of special right triangles: the 45-45-90 triangle and the 30-60-90 triangle. Each of these triangles has specific angles and side ratios that make their properties special.
45-45-90 Triangle
A 45-45-90 triangle, also known as an isosceles right triangle, is a triangle with two congruent angles measuring 45 degrees and one right angle measuring 90 degrees. The side opposite the right angle, known as the hypotenuse, is always the longest side and has a length equal to the square root of 2 times the length of the legs. The two legs of the triangle are congruent and have equal lengths.
- Angle measurements: 45°, 45°, 90°
- Side ratios: 1:1:√2
- Properties: Isosceles, right angle triangle
30-60-90 Triangle
A 30-60-90 triangle is a triangle with angles measuring 30 degrees, 60 degrees, and 90 degrees. The side opposite the 30-degree angle is half the length of the hypotenuse, while the side opposite the 60-degree angle is equal to the square root of 3 times the length of the shorter leg. The hypotenuse is always the longest side.
- Angle measurements: 30°, 60°, 90°
- Side ratios: 1:√3:2
- Properties: Right angle triangle
Special right triangles have a range of applications in geometry, trigonometry, and other branches of mathematics. Their unique properties and ratios make them useful in solving various types of problems and calculations. Understanding the definitions and properties of special right triangles is important for mastering geometry and related concepts.
Properties of Special Right Triangles
A special right triangle is a right triangle that has properties and relationships that make it unique. There are two types of special right triangles: the 45-45-90 triangle and the 30-60-90 triangle. These triangles have specific angles and side lengths that allow for easy calculations and problem solving.
The 45-45-90 triangle, also known as an isosceles right triangle, is characterized by two equal angles of 45 degrees and a right angle of 90 degrees. The sides of this triangle have a ratio of 1:1:√2. The two legs of the triangle are congruent, meaning they have the same length, and the hypotenuse is √2 times the length of each leg.
The 30-60-90 triangle, also known as an equilateral right triangle, is characterized by angles of 30 degrees, 60 degrees, and 90 degrees. The sides of this triangle have a ratio of 1:√3:2. The shortest side, opposite the 30 degree angle, is half the length of the hypotenuse. The longer leg, opposite the 60 degree angle, is √3 times the length of the shortest side. And the hypotenuse, opposite the 90 degree angle, is twice the length of the shortest side.
These special right triangles can be used to solve a variety of mathematical problems, such as finding missing side lengths or angles. By understanding the properties and relationships of these triangles, students can simplify calculations and make problem solving more efficient. They are also helpful in geometric constructions and proofs.
Key Properties of Special Right Triangles:
- 45-45-90 triangle: equal angles of 45 degrees, side lengths in ratio of 1:1:√2
- 30-60-90 triangle: angles of 30 degrees, 60 degrees, and 90 degrees, side lengths in ratio of 1:√3:2
- Legs of 45-45-90 triangle are congruent
- Hypotenuse of 45-45-90 triangle is √2 times the length of the legs
- Shortest side of 30-60-90 triangle is half the length of the hypotenuse
- Longer leg of 30-60-90 triangle is √3 times the length of the shortest side
- Hypotenuse of 30-60-90 triangle is twice the length of the shortest side
Understanding and utilizing the properties of these special right triangles can greatly simplify calculations and problem solving in geometry and other mathematical disciplines. These triangles provide a foundation for advanced concepts and allow for more efficient problem solving techniques.
How to Identify Special Right Triangles
Special right triangles are a specific type of right triangle that have unique characteristics. By understanding these characteristics, it becomes easier to identify special right triangles and apply their properties to solve problems.
There are two main types of special right triangles: the 45-45-90 triangle and the 30-60-90 triangle. These triangles are special because their angle measurements and side ratios have predictable patterns.
45-45-90 Triangle:
- This type of triangle has two equal angles, each measuring 45 degrees.
- The ratio of the lengths of its sides is 1:1:√2.
- The shorter sides are congruent, and the longer side (hypotenuse) is √2 times the length of the shorter sides.
- To identify a 45-45-90 triangle, look for a triangle with two equal angles or sides.
30-60-90 Triangle:
- This type of triangle has angle measurements of 30 degrees, 60 degrees, and 90 degrees.
- The ratio of the lengths of its sides is 1:√3:2.
- The shortest side is half the length of the hypotenuse, and the longer side (opposite the 60 degree angle) is √3 times the length of the shortest side.
- To identify a 30-60-90 triangle, look for a triangle with a 90 degree angle and angles measuring 30 degrees and 60 degrees, or look for specific side ratios.
Identifying these special right triangles can be useful in many applications of geometry and trigonometry. By recognizing these patterns, you can quickly determine side lengths, angles, and other properties of these triangles, making problem-solving easier and more efficient.
Using Special Right Triangles to Solve Problems
Special right triangles, such as the 45-45-90 triangle and the 30-60-90 triangle, provide us with valuable tools for solving various geometric and trigonometric problems. By understanding the relationships between the angles and sides of these triangles, we can determine missing values and solve complex problems with ease.
When dealing with a 45-45-90 triangle, we know that the two legs are congruent, and the hypotenuse is equal to the length of the leg times the square root of 2. This knowledge allows us to find missing side lengths or calculate the measures of angles.
Similarly, a 30-60-90 triangle has specific ratios between its sides. The side opposite the 30-degree angle is half the length of the hypotenuse, and the side opposite the 60-degree angle is equal to the length of the shorter leg times the square root of 3. These ratios can help us solve a variety of problems involving these triangles.
By applying these formulas and understanding the relationships between the angles and sides of special right triangles, we can solve problems involving distances, areas, and angles. These triangles also have practical applications in fields such as engineering, architecture, and physics, where precise calculations and measurements are crucial.
Overall, special right triangles provide us with powerful tools for problem-solving in geometry and trigonometry. By familiarizing ourselves with these triangles and their properties, we can approach complex problems with confidence and find efficient solutions.