Level 1: Simplifying (Reducing) Radicals: 1 1 = 1 = 2 2 = 4 = 3 3 = 9 = 4 4 = 16 = 5 5 = 25 = 6 6 = 36 = 7 7 = 49 =
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1 Name Period Date Unit 5:Special Right Triangles and TrigonometryNotes Packet #1 Section 7.2/7.3: Radicals, Pythagorean Theorem, Special Right Triangles (PA) CRS NCP Work with squares and square roots of numbers PPF Recognize Pythagorean Triples PPF Use the Pythagorean Theorem PPF 28-32Apply properties of , , similar, and congruent triangles CCSS G-SRT.8Use the Pythagorean Theorem to solve right triangles in applied problems GR Use the Distance Formula Simplify radical expressions Level 1: Add, subtract, multiply and divide radicals Rationalize the denominator of a radical expression Perfect Squares and Radicals (Square Roots): The product of a number times itself is called a perfect square number. The square root of a perfect square is a rational number. Find the first 7 perfect square numbers. 1 1 = 1 = 2 2 = 4 = 3 3 = 9 = 4 4 = 16 = 5 5 = 25 = 6 6 = 36 = 7 7 = 49 = Find the square root of each perfect square. Simplifying (Reducing) Radicals: 1. Find the largest perfect square that will divide the number. 2. Rewrite the radical as the product of two radicals. 3. Reduce the perfect square radical. Multiply if there is a coefficient. a) 12 = b) 30 = c) 24 = d) e) f) 1
2 Addition/Subtraction Rule for Radicals: Only like radicals can be added or subtracted. Add or Subtract by combining the coefficients - the radical remains the same. a a 2 a 4 a 2 a 2 a a) b) c) d) = e) = f) = Product Rule for Radicals: For nonnegative real numbers a and b, The product of two radicals is the radical of the product. a b = a b (always reduce your final answer) a) 6 11 = b) 3 15 = c) = d) 3( 2-5 ) = e) 2 ( ) = 2
3 Quotient Rule for Radicals: Quotients - A fraction that contains a radical in the numerator. 1. Reduce radical. 2. Reduce fraction. DO NOT REDUCE a regular number with a radical number. a) = b) = c) = Rationalizing the Denominator: 1. Reduce the fraction if possible. 2. Multiply the denominator and numerator by the radical in the denominator. 2. Simplify the quotient. a) b) c) d) = e) = f) = Level 1 Assignment Radicals Practice Worksheet Page 356 #
4 Derive the Pythagorean Theorem Level 2 Use the Pythagorean Theorem to find missing side lengths Use the Converse of the Pythagorean Theorem to verify a triangle is a right triangle Determine whether the sides of a triangle form a Pythagorean Triple Level 2 *Activity (Pg 349) : Developing the Pythagorean Theorem Pythagorean Theorem: If ΔABC is a triangle, then. What does a, b, and c represent? Examples: Find x. Leave your answer in simplified radical form
5 5. 6. x 7. The bottom end of a ramp at a warehouse is 10 feet from the base of the main dock and is 12 feet long. How high is the dock? Converse of the Pythagorean Theorem: In ΔABC if, then ΔABC is a triangle. Pythagorean Triple: 5
6 Determine whether each set of measures can be the measures of the sides of a right triangle. Then state whether they form a Pythagorean Triple , 40, , 8, , 12, 13 Level 2 Assignment Page #1,2,4-6,11-28,36,37,40-43,46 Level 3 Use the properties of triangles to find missing side lengths Types of Special Triangles: Properties of Triangles: Each leg of a triangle is n units long. The hypotenuseis. How can the Pythagorean Theorem be used to verify that the hypotenuse of any triangle is of its leg? (Draw a diagonal of a square. The two triangles formed are isosceles right triangles). times the length 6
7 Examples: Finding missing side lengths in Find the unknown lengths for each diagram below. Give exact answers Finding the Perimeter/Diagonal of a Square: 1. Find the perimeter of a square with diagonal 12 cm. Draw a picture. 2. Find the diagonal of a square with perimeter 28 m. Draw a picture. Level 3 Assignment Special Triangles Worksheet Page 360 #4,5,11-13,17,22,25,40 7
8 Level 4 Use the properties of triangles to find missing side lengths Types of Special Triangles: Properties of Triangles: The longer leg of a triangle is units long. The shorter leg of a triangle is n units long. The hypotenuseis. How can the Pythagorean Theorem be used to verify that the longer leg of any triangle is times the length of its shorter leg? (Draw an altitude from any vertex of an equilateral triangle. Two congruent triangles are formed). Examples: Finding missing side lengths in Find the unknown lengths for each diagram below. Give exact answers
9 4. If, find CD, b and y. 5. Finding the Perimeter/Diagonal of an Equilateral Triangle: 1. The perimeter of an equilateral triangle is 39 centimeters. Find the length of an altitude of the triangle. Draw a picture. 2. An altitude of an equilateral triangle is 8.3 meters. Find the perimeter of the triangle to the nearest tenth of a meter. Draw a picture. Level 4 Assignment Special Triangles Worksheet Page 360 #2,6-8,14-21,23,24,36,43 9
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