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 24-27 Work with squares and square roots of numbers PPF 24-27 Recognize Pythagorean Triples PPF 28-32 Use the Pythagorean Theorem PPF 28-32Apply properties of 30-60 -90, 45-45 -90, similar, and congruent triangles CCSS G-SRT.8Use the Pythagorean Theorem to solve right triangles in applied problems GR 28-32 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
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) 2 + 8 = e) 27-4 3 = f) 12 + 18 = 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) 10 50 = d) 3( 2-5 ) = e) 2 ( 6 + 3 ) = 2
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) 4 18 12 = b) 5 8 40 = c) 6 50 15 = 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) 2 3 3 = e) 6 4 2 = f) 5 5 10 3 = Level 1 Assignment Radicals Practice Worksheet Page 356 #61-70 3
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. 1. 2. 3. 4. 4
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
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. 1. 30, 40, 50 2. 6, 8, 9 3. 5, 12, 13 Level 2 Assignment Page 353-356 #1,2,4-6,11-28,36,37,40-43,46 Level 3 Use the properties of 45-45 -90 triangles to find missing side lengths Types of Special Triangles: Properties of 45-45 -90 Triangles: Each leg of a 45-45 -90 triangle is n units long. The hypotenuseis. How can the Pythagorean Theorem be used to verify that the hypotenuse of any 45-45 -90 triangle is of its leg? (Draw a diagonal of a square. The two triangles formed are isosceles right triangles). times the length 6
Examples: Finding missing side lengths in 45-45 -90 Find the unknown lengths for each diagram below. Give exact answers. 1. 2. 3. 4. 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
Level 4 Use the properties of 30-60 -90 triangles to find missing side lengths Types of Special Triangles: Properties of 30-0 -90 Triangles: The longer leg of a 30-60 -90 triangle is units long. The shorter leg of a 30-60 -90 triangle is n units long. The hypotenuseis. How can the Pythagorean Theorem be used to verify that the longer leg of any 30-60 -90 triangle is times the length of its shorter leg? (Draw an altitude from any vertex of an equilateral triangle. Two congruent 30-60 -90 triangles are formed). Examples: Finding missing side lengths in 30-60 -90 Find the unknown lengths for each diagram below. Give exact answers. 1. 2. 3. 8
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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