Math Example--Ratios and Rates--Example 14

Topic

Ratios and Rates

Description

This example explores the concept of special right triangles using ratios. The image shows a right triangle with sides labeled 3, 3 2 2 ​ , and 6 units. Students are guided through a step-by-step process to verify if this triangle is similar to a 30°-60°-90° triangle.

Understanding special right triangles and their ratios is crucial in trigonometry and geometry. This example demonstrates how ratios can be used to compare the sides of a given triangle to a standard 30°-60°-90° triangle. By scaling the sides of the standard triangle (1 : √2 : 2) and comparing them to the given triangle, students can see how ratios maintain proportionality in similar shapes, even with irrational numbers.

Providing multiple worked-out examples is essential for students to fully grasp 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 more complex geometric problems.

Teacher's Script: Let's look at this right triangle. Can you identify the lengths of its sides? Now, we want to check if this triangle is similar to a 30°-60°-90° triangle. What are the side ratios of a standard 30°-60°-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 3? How does this compare to our given triangle?

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