Math Example--Ratios and Rates--Example 15

Topic

Ratios and Rates

Description

This example focuses on verifying the similarity of a given triangle to a 45°-45°-90° triangle using ratios. The image shows a right triangle with sides labeled 4, 4, and 4√2 ​ units. Students are guided through the process of comparing this triangle's side ratios to those of a standard 45°-45°-90° triangle.

Understanding special right triangles and their ratios is a fundamental concept in trigonometry and geometry. This example illustrates how ratios can be used to compare the sides of a given triangle to a standard 45°-45°-90° triangle. By scaling the sides of the standard triangle (1 : 1 : √2 ​ ) and comparing them to the given triangle, students can see how ratios maintain proportionality in similar shapes, even with irrational numbers.

Exposure to multiple worked-out examples is crucial for students to fully comprehend the concept of ratios and their applications in geometry. Each example in this series presents a unique scenario, allowing students to practice identifying ratios, scaling them, and recognizing similarity in special right triangles. This repetition helps reinforce the concept and builds confidence in applying ratio skills to various geometric problems.

Teacher's Script: Let's examine this right triangle. Can you identify the lengths of its sides? Now, we want to check if this triangle is similar to a 45°-45°-90° triangle. What are the side ratios of a standard 45°-45°-90° triangle? How can we use these ratios to determine similarity with our given triangle? Remember, similar triangles have proportional sides. What happens if we multiply each side of the standard triangle by 4? How does this compare to our given triangle? Can you explain why this confirms similarity?

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