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Understanding the Pythagorean Theorem and Its Converse in Right Triangles

This chapter explores the Pythagorean Theorem and its converse, essential concepts in geometry related to right triangles. The Pythagorean Theorem states that in a right triangle, the sum of the squares of the legs equals the square of the hypotenuse (a² + b² = c²). Exercise examples delve into practical applications, such as determining heights using the theorem. Additionally, the converse helps classify triangles as acute, right, or obtuse based on side lengths, enhancing understanding of triangle properties.

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Understanding the Pythagorean Theorem and Its Converse in Right Triangles

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  1. Chapter 7.1 & 7.2 Notes: The Pythagorean Theorem and its Converse Goal: To use the Pythagorean Theorem and its Converse.

  2. Right Triangles: • In a right triangle, the side opposite the right angle is the longest side, called the hypotenuse. The other two sides are the legs of a right triangle. • Theorem 7.1 Pythagorean Theorem: In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. a2 + b2 = c2

  3. Find the value of x. Leave your answer in simplest radical form. Ex.1: Ex.2: Ex.3: A 16-foot ladder rests against the side of the house, and the base of the ladder is 4 feet away. Approximately how high above the ground is the top of the ladder?

  4. When the lengths of the sides of a right triangle are integers, the integers form a Pythagorean Triple. • Common Pythagorean Triples and Some of Their Multiples:

  5. Ex.4: Find the area of the isosceles triangle with side lengths 10 meters, 13 meters, and 13 meters. • Find the value of x. Leave your answer in simplified radical form. Ex.5: Ex.6:

  6. Theorem 7.2 Converse of the Pythagorean Theorem: If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. If c2 = a2 + b2, then ∆ABC is a right triangle.

  7. Tell whether a triangle with the given side lengths is a right triangle. Ex.7: 5, 6, Ex.8: 10, 11, 14 Ex.9: 8 4

  8. The Converse of the Pythagorean Theorem is used to determine if a triangle is a right triangle, acute triangle, or obtuse triangle. • If c2 = a2 + b2, then the triangle is a right triangle. • If c2 > a2 + b2, then the triangle is an obtuse triangle. • If c2 < a2 + b2, then the triangle is an acute triangle.

  9. Determine if the side lengths form a triangle. If so, classify the triangle as acute, right, or obtuse. Ex.10: 15, 20, and 36 Ex.11: 6, 11, and 14 Ex.12: 8, 10, and 12 Ex.13: 4.3, 5.2, and 6.1

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