Algebra 1B. Unit 9. Algebraic Roots and Radicals. Student Reading Guide. and. Practice Problems

Transcription

1 Name: Date: Period: Algebra 1B Unit 9 Algebraic Roots and Radicals Student Reading Guide and Practice Problems Contents Page Number Lesson 1: Simplifying Non-Perfect Square Radicands 2 Lesson 2: Radical Expressions Containing Variables 4 Lesson 3: Simplifying Roots of Variables 6 Lesson 4: Operations with Radicals 8 Lesson 5: Pythagorean Theorem 12 Lesson 6: Distance 16 Lesson 7: Trigonometric Ratios 18 Lesson 8: Solving Right Triangles 21 The main thing you ll learn in this Unit: What does it mean to write a radical in simplest terms? 1

2 Lesson 9.1: Simplifying Non-Perfect Square Radicands Steps: Example: 125 = 25 5 = 5 5 When simplified form still contains a radical, it is said to be irrational. Examples Practice 72 = 18 = 360 = 24 = 80 = 396 = Which of the following does not have an irrational simplified form? 40 = 60 = 80 = 100 = Simplify: (circle A, B, C, or D) Given: A B C D 5 20 = = = =

3 When 5 20 is written in simplest radical form, the result is k 5. What is the value of k? a) 20 b) 10 c) 7 d) 4 When 72 is expressed in simplest a b. For m, what is the value of a? a) 6 b) 2 c) 3 d) 8 Express 3 48 in simplest radical form. 9.1 Homework

4 Lesson 9.2: Radical Expressions Containing Variables Rule: Square Roots of Variables Examples Practice x 12 = x 24 = d 4 = a 16 = Simplify: (circle A, B, C, or D) Given: A B C D b 16 = b 14 b 8 b 4 No real solution b 6 = b 4 b 3 b 2 No real solution m 24 = m 12 m 22 m 2 No real solution h 14 = h 7 h 12 h 5 No real solution 16 x 8 = x 6 x 4 x 2 No real solution 4

5 9.2 Homework: Simplify 1. b 7 9. x 7 2. b x 5 3. b x 4 4. b x 8 5. b x 6 6. b x 3 7. b x 9 8. b 9 5

6 Lesson 9.3 Simplifying Roots of Variables Simplifying Roots of Variables Rule: Simplify Examples Practice 50x 4 y 12 z 3 = 8x 5 y 6 z 4 = m 11 n 22 = 16x 7 y 4 z = Simplify: (circle A, B, C, or D) Given: A B C 20x 3 2x 5x 2x 5x 2 4x 2 5x 25x 62 x 30 25x 5x x 30 5x 31 50m 12 n 14 p 16 5m 6 n 7 p m 5 n 7 p 8 5m 6 n 3 p 8 2 6

7 9.3 Homework 1. 40x 5 y 8 z x 4 y 3 z 5 2. x 6 y 7 z x 2 y 8 z x 4 y 6 z x 9 y 7 z x 6 y 5 z x 3 y 6 z x 3 y 2 z x 5 y 4 z x 4 y 5 z x 5 y 9 z 15 7

8 Lesson 9.4: Operations with Radicals Adding or Subtracting Radicals Radicals can be added and subtracted when they have like terms. Like Terms means they have. Identify the like terms Examples Practice 4 6, , , , 11 Rule: Hint: Examples Practice = = = = = = Radicals must be simplified before adding or subtracting Examples Practice 12 3 = = = = = 8

9 Multiplying Radicals Rule: Hint: Examples Practice = = = = = = Multiplying Radicals After multiplying, check to see if radicand can be simplified Examples Practice = = = 4 x 2 3 y 3 = Multiplying Polynomials Involving Radicals F O I L Hint: 9

10 Examples Rationalizing the Denominator Which of these expressions has a rational denominator? Rule: Hint: Examples Practice 12 = = 46 = 3 5 = = 4 5 = 10

11 9.4 Homework

12 Lesson 9.5: Pythagorean Theorem A right triangle is a triangle with a right angle. The sides that form the right angle are the legs. The side opposite the right angle is the hypotenuse. The hypotenuse is also the longest side. leg hypotenuse leg Rule: Pythagorean Theorem (R1) Hint: Is the missing side a leg or the hypotenuse of the right triangle? Find the length of the missing side of the right triangle. Example: Practice: x x 20 x 6 9 x The dimensions of a high school basketball court are 84' long and 50' wide. What is the length of from one corner of the court to the opposite corner? 12

13 Pythagorean Theorem Applications: The Pythagorean Theorem can also be used in figures that contain right angles. Example: Find the perimeter and area of the square. Before finding the perimeter of the square, we need to first find the length of each side. Remember, in a square all sides are congruent. 18 cm s P sq = 4s A sq = s 2 Example: Find the area of the triangle. The base of the triangle is given, but we need to find the height of the triangle. By definition, the altitude (or height) of an isosceles triangle is the perpendicular bisector of the base. 13 ft h 13 ft A = 1 2 bh 5 ft 5 ft 10 ft Example: Find the perimeter and area of the rectangle. 10 in 8 in P rect = 2l + 2w A rect = lw Converse of the Pythagorean Theorem (R2): If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. B Rule: a c Example: Tell whether the triangle is a right triangle. C b A D 24 E 25 7 F 13

14 Theorem (R3): If the square of the longest side of a triangle is greater than the sum of the squares of the other two sides, then the triangle is obtuse. Rule: Theorem (R4): If the square of the longest side of a triangle is less than the sum of the squares of the other two sides, then the triangle is acute. Rule: Example: Classify the triangle as acute, right, or obtuse. Example: Tell whether 12, 3, 3 15 represent the sides of an acute, right, or obtuse triangle. Example: Classify the triangle is acute, right, obtuse, or not a triangle. Example: Classify the triangle is acute, right, obtuse, or not a triangle. Review If c 2 = a 2 + b 2, then triangle is right. If c 2 > a 2 + b 2, then triangle is obtuse. If c 2 < a 2 + b 2, then triangle is acute. 14

15 9.5 Homework: 1. In the triangle above the hypotenuse is side, the legs are sides and. Refer the diagram below to answer questions 2-7. Write the answer in simplest radical form. 2. If AB = 33 and BC = 56, then AC =. 3. If AB = 12 and AC = 37, then BC =. 4. If BC = 80 and AC = 89, then AB =. 5. If AC = 14 and AB = 5 2, then BC =. 6. If BC = 9 and AC = 16, then AB =. 7. If AB = 5 2 and BC = 7 6, then AC =. Classify the sides as lengths of an acute, right, obtuse, or not a triangle , 7, , 4, , 14, , 4 7, Find the perimeter and area of the triangle. 15

16 Lesson 9.6: Distance Computing the distance between two points in the plane is an application of the Pythagorean Theorem for right triangles. Computing distances between points in the plane is equivalent to finding the length of the hypotenuse of a right triangle. Relationship between the Pythagorean Theorem & Distance Formula The Pythagorean Theorem states a relationship among the sides of a right triangle. The distance formula calculates the distance using point's coordinates. The Pythagorean Theorem is true for all right triangles. If we know the lengths of two sides of a right triangle then we know the length of the third side. The distance between two points, whether on a line or in a coordinate plane, is computed using the distance formula. The Distance Formula: The distance 'd' between any two points with coordinates x 1, y 1 and x 2, y 2 and is given by the formula: Note: Recall that all coordinates are (x coordinate, y coordinate). Example: Calculate the distance from Point K to Point I. Example: Calculate the distance from Point J to Point K. Example: Calculate the distance from Point J to Point K. Example: Calculate the distance from Point H to Point K. Example: Calculate the distance from Point G to Point K. Example: Calculate the distance from Point I to Point H. Example: Calculate the distance from Point G to Point H. 16

17 9.6 Homework Find the distance between the given points. For irrational answers, write in simplest terms. 47.(4,-2) and (9,10) 48.(7,2) and (3,12) 49.(-1, 9) and (8, -4) 50.(3,2) and (4, -1) 51.(3, 10) and (-3, 2) 17

18 Lesson 9.7: Trigonometric Ratios Trigonometry is a branch of mathematics that deals with relationship of the sides and angles of triangles. A trigonometric ratio is the ratio of the two lengths of a right triangle. There are 3 ratios for each acute angle of a right triangle. The ratios are called sine, cosine, and tangent abbreviated sin, cos, and tan respectively. Trig Function Trig Formula Hint Sine sin θ = Cosine cos θ = Tangent tan θ = SOHCAHTOA B In each right triangle, there are 2 acute angles. In the triangle to the left <A and <B are the acute angles. a c Find the side opposite <A, side adjacent <A, hypotenuse. side opposite <A = C b A side adjacent <A = hypotenuse = SOHCAHTOA Sine sin A = side opposi te <A hypotenuse = Cosine cos θ = side adjacent <A hypotenuse = Tangent tan θ = side opposite <A side adjacent = Example: Find the sin, cos, and tan of <F. What is the side opposite, side adjacent, and the hypotenuse of the right triangle? 18

19 Example: What is the side opposite to <J? What is the hypotenuse of the triangle? What is the side adjacent to <J? Example: What is the sinr? What is the cosr? What is the tanr? Example: Evaluate sin60. Round to the nearest tenthousandth. Example: Evaluate cos45. Round to the nearest tenthousandth. Example: Evaluate tan30. Round to the nearest tenthousandth. Find the length of LM. Find the length of LP. 19

20 9.7 Homework Refer to the triangle below to answer questions The side opposite to <W is. 2. The side adjacent to <W is. 3. The hypotenuse is. 4. The sinb =. 5. The cosb =. 6. The tanb =. Evaluate. Round the nearest ten-thousandth. 7. sin45 o 8. tan 77 o 9. cos69 o 10. tan33 o 11. Find the length of RF. Round to the nearest hundredth. 12. Find the length of SC. Round to the nearest hundredth. 13. Find the length of BK. Round to the nearest hundredth. 20

21 Lesson 9.8: Solving Right Triangles To solve a right triangle means to find the length of each side and the measure of each angle in the triangle. When using trigonometric ratios to solve a right triangle, you need to know either the length of 2 sides or the length of one side and the measure of one the acute angles. Remember: m < A + m < B + m < C = 180 a 2 + b 2 = c 2 SOHCAHTOA Inverses Example: Find sin Round to the nearest hundredth. Example: Find tan Round to the nearest hundredth. Example: Find cos Round to the nearest hundredth. In ΔABC, find the m<a, m<c and BC. m<a = m<c = BC = Find CE. Find m<c. Find the m<e. 21

22 9.8 Homework Solve the triangle. Round to the nearest hundredth. m<c = AC = BC = m<d = m<f = DF = m<g = m<i = GH = m<j = KL = JL = 22