Exploring Applications of the Pythagorean Theorem

Objective:

Students will be able to apply the Pythagorean Theorem to find the height of an isosceles triangle, prove the distance formula, and classify a triangle as acute, right, or obtuse.

Assessment:

Students will be assessed through a worksheet that includes problems requiring them to apply the Pythagorean Theorem to find the height of an isosceles triangle, prove the distance formula, and classify triangles.

Key Points:

• Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two other sides (a^2 + b^2 = c^2).

• Height of an Isosceles Triangle: To find the height of an isosceles triangle, the Pythagorean Theorem can be used to relate the base and height of the triangle.

• Distance Formula: The distance formula is derived from the Pythagorean Theorem and is used to find the distance between two points in a coordinate system.

• Triangle Classification: Triangles can be classified as acute (all angles < 90 degrees), right (one angle = 90 degrees), or obtuse (one angle > 90 degrees) based on their side lengths.

Opening:

• Engage students by asking, "How can we use the Pythagorean Theorem in real-life situations?"

• Show a short video or image of a triangle to spark curiosity

Introduction to New Material:

• Discuss the Pythagorean Theorem and its formula

• Explain how to find the height of an isosceles triangle using the Pythagorean Theorem

• Anticipate common misconception: Students might confuse the legs and hypotenuse of a triangle

Guided Practice:

• Provide examples for finding the height of an isosceles triangle step by step

• Scaffold questions starting with basic calculations leading to more complex problems

• Monitor student performance by circulating the classroom and offering guidance as needed

Independent Practice:

• Assign problems requiring students to apply the Pythagorean Theorem to find the height of different isosceles triangles

• Students will also prove the distance formula and classify triangles independently

• Encourage students to show all their work and explanations clearly

Closing:

• Have students share their findings and explain how they used the Pythagorean Theorem

• Summarize the key points covered during the lesson

Extension Activity:

• Create a challenge for students to find the height of irregular triangles using the Pythagorean Theorem

Homework:

• Students are required to complete a set of problems similar to the ones covered in class, focusing on applying the Pythagorean Theorem to different triangle scenarios.

Standards Addressed:

• CCSS.MATH.CONTENT.HSG.SRT.C.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

• CCSS.MATH.CONTENT.HSG.SRT.C.8b: Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.