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Simplifying Radicals & Pythagorean Theorem

Simplifying Radicals & Pythagorean Theorem. Notes 25 – Sections 0.9 & 8.2. Essential Learnings. Students will understand and be able to simplify radical expressions. Students will understand and be able to use the Pythagorean Theorem to solve problems involving right triangles.

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Simplifying Radicals & Pythagorean Theorem

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  1. Simplifying Radicals & Pythagorean Theorem Notes 25 – Sections 0.9 & 8.2

  2. Essential Learnings • Students will understand and be able to simplify radical expressions. • Students will understand and be able to use the Pythagorean Theorem to solve problems involving right triangles.

  3. Simplifying Radicals • A radical expression is an expression that contains a square root. The radicand is the part under the square root. Radicand

  4. Simplifying Radicals For a radical to be in simplest form: • No radicands have a perfect square factor (other than 1). • No radicands contain fractions. • No radicals appear in the denominator of a fraction.

  5. Product Property • For any two numbers a and b such that • Example 1:

  6. Quotient Property • For any two numbers a and b such that and • Example 2:

  7. Example 3 Simplify each radical. a) b)

  8. Example 4 Simplify the radical.

  9. Example 5 Simplify the radical.

  10. Pythagorean Theorem In a right triangle with legs a and b and hypotenuse c: c a b

  11. Example 1 Find x. x 4 7

  12. Example 2 Find x. 12 x 8

  13. Pythagorean Triples • A set of three nonzero whole numbers that make the Pythagorean Theorem true. 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25

  14. Example 3 Use a Pythagorean triple to find x. 26 24 x

  15. Converse of the Pythagorean Theorem If the sum of the squares of the lengths of the shortest sides of a triangle is equal to the square of the length of the longest side, then the triangle is a right triangle. c a b

  16. Pythagorean Inequality Theorem If c2 < a2 + b2 , then ΔABC is acute. C b a A B c

  17. Pythagorean Inequality Theorem If c2 > a2 + b2 , then ΔABC is obtuse. B c a A C b

  18. Triangle Inequality Theorem • The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. a + b > c b + c > a a + c > b C b a A B c

  19. Example 4 Determine whether the set of numbers can be measures of the sides of a triangle. If so, classify as acute, right, or obtuse. 10, 11, 13

  20. Example 5 Determine whether the set of numbers can be measure of the sides of a triangle. If so, classify as acute, right, or obtuse. 10, 12, 23

  21. Example 6 Determine whether ΔXYZ is an acute, right, or obtuse triangle for the given vertices. X (-3, -2), Y (-1, 0), Z (0, -1)

  22. Assignment Simplifying Radicals WS p. 545: 1-3, 5-12, 15-20, 30, 31 Unit Study Guide 4 #1-9 graded on Wednesday

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