Exploring 45-45-90 Right Triangles

Objective:

Students will be able to understand the properties of 45-45-90 right triangles, identify them in geometric figures, and apply the 45-45-90 theorem to solve problems.

Assessment:

Students will be assessed through a worksheet containing various questions related to identifying and solving problems involving 45-45-90 right triangles.

Key Points:

• Definition of an isosceles right triangle: An isosceles right triangle is a triangle with a ninety degree angle and exactly two sides that are the same length. This triangle is also called a 45-45-90 triangle.

• Properties of a 45-45-90 triangle: In a 45-45-90 triangle, the two legs (sides adjacent to the right angle) are congruent, and the length of the hypotenuse is √2 times the length of each leg.

• 45-45-90 Theorem: In a 45-45-90 triangle, if the length of each leg is "x", then the length of the hypotenuse is x√2.

• Understanding the 45-45-90 theorem: The 45-45-90 theorem helps us relate the side lengths in an isosceles right triangle and provides a shortcut to find the length of the hypotenuse.

• Identifying 45-45-90 right triangles in different figures

• Applying the theorem to solve problems

Opening:

• Engage students by presenting a visual of a 45-45-90 right triangle and asking: "What do you notice about the sides of this triangle?"

• Discuss responses as a class to activate prior knowledge.

Introduction to New Material:

• Explain the definition of an isosceles right triangle and introduce the properties of a 45-45-90 triangle.

• Present the 45-45-90 theorem and discuss its significance.

• Anticipate misconception: Students may confuse the 45-45-90 theorem with the Pythagorean theorem.

Guided Practice:

• Provide examples of 45-45-90 right triangles for students to identify and discuss the properties.

• Scaffold questioning from easy (identifying the hypotenuse) to harder (solving for side lengths).

• Monitor student understanding through guided questions and peer discussions.

Independent Practice:

• Assign a worksheet with a mix of problems involving 45-45-90 triangles.

• Students will have to apply the 45-45-90 theorem to solve for missing side lengths.

• Encourage students to show all work and justify their answers.

Closing:

• Summarize key points about 45-45-90 triangles as a class.

• Have students explain how the 45-45-90 theorem can be useful in solving real-world problems involving right triangles.

Extension Activity:

• Challenge early finishers to explore other types of special triangles (e.g., equilateral triangles, 30-60-90 triangles) and compare their properties with 45-45-90 triangles.

Homework:

• Homework: Create a set of 5 problems for students to practice identifying and applying the 45-45-90 theorem to solve for missing sides in triangles.

Standards Addressed:

• CCSS.MATH.CONTENT.HSG.SRT.B.4: Use the properties of an isosceles right triangle to find missing side lengths. (x2)