Geometry- Unit 6 Notes. Simplifying Radicals

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1 Geometry- Unit 6 Notes Name: Review: Evaluate the following WITHOUT a calculator. a) 2 2 b) 3 2 c) 4 2 d) 5 2 e) 6 2 f) 7 2 g) 8 2 h) 9 2 i) 10 2 j) 2 2 k) ( 2) 2 l) 2 0 Simplifying Radicals n r Example 1: What does 36 mean? Example 2: What does 3 27 mean? How to simplify radicals: Example 3: Break the radicand into 2. Find groups of (index) 3. Groups, any number not in a group Example 4: 100 Example 5: 33 Example 6: 2 50

2 Your Turn! 1) 45 2) 75 3) 64 4) 22 Example 7: Explain what 121 = 11 means? The 121 = 11 means that the of 121 is 11 or is 121. Geometry- Unit 6 Notes Review: With your table partner, match the radical with the simplified version. Radical 9 Simplified Name: /9.2 Pythagorean Theorem Right Triangle Pythagorean Theorem: In a right triangle, the sum of the squares of the length of the legs equals the OR + =

3 Converse of Pythagorean Theorem: If the lengths of the three sides of a triangle satisfy the Pythagorean equation, then the triangle. Pythagorean Triples: 3, 4,5 5, 12, 13 7, 24, 25 8, 15, 17 Example 1: Find the value of c. Example 2: Find the value of a Example 3: Find the length of the diagonal. a. 30 m, 40 m, 50 m Example 4: Which of the following are the side lengths of a right triangle? b. 10 in, 20 in, 24 in c. 12 cm, 18 cm, 22 cm d. 130 ft, 50 ft, 120 ft Your Turn! 1) The legs of a right triangle are 5 inches and 7 inches, find the length of the hypotenuse. Leave answer in simplest radical form.

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7 2) The longest side of a right triangle is 14 cm and one of the legs is 2 cm. Find the length of the other leg. Leave answer in simplest radical form. Geometry- Unit 6 Notes Name: Review: Classify the following triangles by side length and angle measures. Word Bank: ACUTE OBTUSE RIGHT SCALENE ISOSCELES EQUILATERAL - Also known as - Find the length of the hypotenuse Triangles Example 1: Find the missing sides of the triangle. Leave answer in simplest radical form.

8 Example 2: Find the missing sides of the triangle. Leave answer in simplest radical form. Example 3: Find the missing sides of the triangle. Leave answer in simplest radical form. Example 4: Find the length of the diagonal of the square. Leave in simplest radical form. Your Turn! 1) The hypotenuse of an isosceles right triangle is What is the length of the leg? What is the perimeter of the triangle?

9 2) Find the values of a and b. Leave answer in simplest radical form. Geometry- Unit 6 Notes Review: Determine if each of the following statements are TRUE or FALSE TRUE/FALSE The square root of 25 is 5. TRUE/FALSE TRUE/FALSE 9 = 3 The Pythagorean Theorem can be applied to all triangles. Name: TRUE/FALSE In the Pythagorean Theorem, a 2 + b 2 = c 2, c is the longest side of the triangle called the hypotenuse. TRUE/FALSE In an isosceles right triangle, there are two congruent sides and the third side is 2 times the length of the congruent sides. TRUE/FALSE In a triangle, if the hypotenuse is 15, then the legs are Triangles

10 Example 1: Find the value of the two other sides of the triangle. Example 2: Find the value of the two sides of the triangle. Example 3: The equilateral triangle has been divided into 2 triangles. Find the values of g and h. Example 4: Find the value of the other two sides of the triangle.

11 Example 5: Find the value of x and y. Your Turn! 1) The following are triangles. Find the missing values in simplest radical form. a. b. 2) The following are triangles. Find the missing values in simplest radical form. Geometry- Unit 6 Notes Name: Review: For a given angle in a right triangle (not the right angle), the HYPOTENUSE is the longest side of the triangle, opposite the right angle the ADJACENT side is next to the angle, but not the hypotenuse the OPPOSITE side is across from the angle For the angle indicated, identify the opposite, adjacent and hypotenuse sides. Identify for which angle the sides are labeled.

12 Right Triangle Trigonometry What is right triangle trigonometry? Only used for triangles Ratios that compare of a right triangle Three trigonometry ratios: Can use to determine of a right triangle Example 1: Write the trigonometric ratios for angle C the following triangle. s in C = c os C = t an C = Example 2: Write the trigonometric ratios for angle D the following triangle. s in D = c os D = t an D =

13 How to use trigonometry to find side lengths 1. Determine what sides are given and what ratio to use. Example 1: 2. Set up ratio. 3. Create proportion. 4. Cross multiply 5. Solve (Do not use calculator until this step). Round answer to nearest hundredth (2 decimal places) Example 2: Find the missing value. Example 3: Find the missing value. Example 4: Find the missing value. Example 5: A 15 ft ladder leans against a building at a 55 o angle from the ground. How far from the the building is the bottom of the ladder? How to use INVERSE trigonometry to find angle measures 1. Determine what sides are given and what ratio to use. Example 6: 2. Set up ratio. 3. Use inverse trigonometric function (sin -1, cos -1, tan -1 ) 4. Evaluate (don t forget parentheses!) Example 7: Find the value of θ. Round to nearest hundredth.

14 Example 8: Find the value of θ. Round to nearest hundredth. Example 9: Find the value of θ. Round to nearest hundredth. Your Turn! 1) Use INVERSE trigonometry to find the measures of angles A and B. 2) Use trigonometry to find the lengths of AC and BC.