Exploring Inscribed Similar Triangles

Objective:

Students will be able to understand and apply the Inscribed Similar Triangles Theorem by drawing altitudes in right triangles to create similar triangles.

Assessment:

Students will demonstrate their mastery of the Inscribed Similar Triangles Theorem by accurately identifying and explaining the similarities between the triangles in different examples.

Key Points:

• Similarity of Triangles: If two objects are similar, their corresponding angles are congruent and their sides are proportional in length.

• Altitude of a Right Triangle: The altitude of a right triangle creates similar triangles.

• Inscribed Similar Triangles Theorem: If an altitude is drawn from the right angle of any right triangle, then the two triangles formed are similar to the original triangle and all three triangles are similar to each other.

• Formula for Altitude in a Right Triangle: The formula to calculate the altitude of a right triangle is given by:
  Height = Base * Height / Hypotenuse

Opening:

• Engage students by posing the question: "How can we prove that two triangles are similar?"

• Show a visual illustration of a right triangle with altitudes drawn to introduce the concept

Introduction to New Material:

• Explain the concept of inscribed similar triangles

• Discuss how drawing an altitude from the right angle forms similar triangles

• Anticipate misconception: Students may confuse the concept of similar triangles with congruent triangles

Guided Practice:

• Provide examples of right triangles and guide students through drawing altitudes to create similar triangles

• Scaffold questioning from easier examples to more complex ones

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

Independent Practice:

• Instruct students to draw altitudes in different right triangles and determine the similarities between the triangles

• Assign an activity where students have to identify and explain the similarities between the triangles formed

Closing:

• Summarize the key points by asking students to explain how drawing altitudes in a right triangle creates similar triangles

Extension Activity:

• For early finishers, provide a challenge where they have to prove the triangles are similar using specific angle and side measurements

Homework:

• Homework suggestion: Draw three different right triangles and draw altitudes to create similar triangles. Identify the similarities between the original triangle and the new triangles.

Standards Addressed:

• CCSS.MATH.CONTENT.HSG.SRT.B.4: Prove theorems about triangles, including the Inscribed Similar Triangles Theorem

• CCSS.MATH.CONTENT.HSG.SRT.B.4: Understand similarity in terms of similarity transformations