6 From Base to Branch Copyright © 2026 Pearson Education, Inc. From Base to Branch (50 – 60 minutes) Learning Objective(s): Students will apply the Pythagorean Theorem to find missing side lengths in right triangles. Students will use sine, cosine, and tangent ratios to solve for unknown sides or angles in right triangles. Students will recognize and apply side length relationships in 30°– 60°– 90° and 45°– 45°– 90° triangles. Material needed: Student pages: From Base to Branch Calculator Lesson Procedure: Warm–Up 5–10 minutes Prompt: Think about situations in construction, climbing, or sports where you see diagonal structures or cables. Why do you think understanding the angles or lengths involved might be important? Discuss: angles of triangle and right triangles Guided Instruction 10 minutes Present: scenario for From Base to Branch. Example: A zipline tower is 20 feet tall. The zipline connects to the ground 30 feet away from the base. In relation to a triangle, how would you represent the zipline? The zipline is the hypotenuse of the right triangle formed. How could you find the length of the zipline? Use the Pythagorean Theorem: a2 + b2 = c2 How would you find the angle of elevation? Use the trig ratio tan(θ) = opposite/adjacent. Review: key terms – angle of elevation, hypotenuse, right triangle, tangent angle of elevation: angle measured upward from a horizontal line to an object above the line of sight hypotenuse: longest side of a right triangle, opposite the right angle right triangle: triangle that has a 90° angle tangent: trigonometric ratio used in right triangles to compare the opposite side to the adjacent side of a given angle Independent Practice 25 minutes Distribute: student activity From Base to Branch Allow students to work individually or in pairs. Closure 10–15 minutes Review Answers: 1. 17.49 ft 2. 21.80° 3. a. 18.67 ft; b. The rope would have to be longer. 4. a. 45°– 45°– 90° right triangle; b. 14.14 in. 5. a. 8.40 ft; b. The shadow would be longer. 6. 12.12 ft 7. a. 12 ft; b. 10.39 ft 8. a. 47.17 ft; b. 32.01° Discuss: How do trigonometric ratios and the Pythagorean Theorem make measuring and building structures safer and more efficient?
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