1. R ADICAL E XPRESSIONS AND EQUATIONS Chapter 11

2. INTRODUCTION We will look at various properties that are used to simplify radical expressions. We will then apply properties to problems using the Pythagorean Theorem as well as the Distance and Midpoint formulas and solving special triangles. We will solve radical expression and graph square root functions. Finally, solving right triangles we obtain definitions for three key trigonometric functions which can then be used to solve real world problems such as problems involving angles of elevation and depression.

3. S IMPLIFYING RADICALS (11.1) Radical expressions : Algebraic expressions that contain the radical sign. We can simplify a radical expression by removing perfect-square factors from the radicand. We can use the Multiplication Property of Square Roots to help in the simplification of radical expressions.

4. S IMPLIFYING RADICALS (11.1) We can rewrite the radicand as a product of the perfect-square factors times the remaining factors by using the Multiplication Property of Square Roots.

5. S IMPLIFYING RADICALS (11.1) Simplifying radical expressions that contain variables is also possible. A variable with a non-zero, even exponent is a perfect square. A variable with an odd exponent (other than 1 and -1) are the product of a perfect square and the variable.

6. S IMPLIFYING RADICALS (11.1)

7. We can use the Division Property of Square Roots to simplify expressions. When the denominator of the radicand is a perfect square, it is easier to simplify the numerator and denominator separately. When the denominator is not a perfect square, it maybe easier to divide first and then simplify the radical expression.

8. S IMPLIFYING RADICALS (11.1)

9. When a radicand in the denominator of a radical expression is not a perfect square, we may have to rationalize the denominator to simplify the expression. To rationalize a radical expression we multiply the numerator and the denominator by the same radical expression. The radical expression we choose should make the denominator a perfect square. Since we are multiplying the numerator and the denominator by the same radical expression, we are essentially multiplying by one.

10. S IMPLIFYING RADICALS (11.1)

11. The summary below can help you determine whether a radical expression is in the simplest radical form: The radicand has no perfect-square factors other than 1. The radicand has no fractions. The denominator of a fraction has no radical.

12. T HE PYTHAGOREAN THEOREM (11.2) A right triangle is a triangle that has one angle that is 90 o degrees. The Pythagorean Theorem describes the relationship between the lengths of the sides of the right triangle. hypotenuse, c leg, a leg, b

13. T HE PYTHAGOREAN THEOREM (11.2) The Pythagorean Theorem can be used to solve for problems involving right triangles and their sides. The Pythagorean Theorem In any right triangle, the sum of the squares of the length of the legs is equal to the square of the length of the hypotenuse. a 2 + b 2 = c 2

14. T HE PYTHAGOREAN THEOREM (11.2) Sample Problem What is the length of the hypotenuse of the right triangle at the right? c 12 cm 9 cm

15. T HE PYTHAGOREAN THEOREM (11.2) We can use the Pythagorean Theorem to find the length of one of the legs if we know the lengths of the hypotenuse and the other leg. Sample Problem A fire truck parks beside a building such that the base of the ladder is 16 ft. from the building. The fire truck extends its ladder 30 ft. The ladder sits on top of the truck which is 10 ft. above the ground. How high is the top of the ladder from the ground?

16. T HE PYTHAGOREAN THEOREM (11.2) Not all converse statements are true, but the converse of the Pythagorean Theorem is always true. Thus, we can use this converse to determine if a given triangle is right triangle or if it is not a right triangle. The Converse of the Pythagorean Theorem If a triangle has sides of lengths a, b, and c, and a 2 + b 2 = c 2, then the triangle is a right triangle with hypotenuse of length c.

17. T HE PYTHAGOREAN THEOREM (11.2) Sample Problem Determine whether the given lengths can be sides of a right triangle. a) 5 in., 12 in., 13 in. b) 7 m, 9 m, 12 m

18. T HE PYTHAGOREAN THEOREM (11.2) The Pythagorean Theorem and its converse has many uses in solving problems in physics. One use is in solving for problems involving forces. Sample Problem If two forces pull at right angles to each other, the resultant force is represented as the diagonal of a rectangle. The diagonal forms a right triangle with two of the perpendicular side of the rectangle. For a 30-lb force and a 40-lb force, the resultant force is 50-lbs. Are the forces pulling at right angles to each other?

19. T HE DISTANCE AND MIDPOINT FORMULAS (11.3) For vertical and horizontal line segments, we can find their lengths by subtracting their y- and x- coordinates, respectively. For any two points P(x 1, y 1 ) and Q(x 2, y 2 ) not on a horizontal or vertical line, we can graph the points and form a right triangle. We can then use the Pythagorean Theorem to find the distance between the points. Let’s take a look at how this is done.

20. T HE DISTANCE AND MIDPOINT FORMULAS (11.3)

21. We can find an exact value by substituting values into the Distance Formula and simplifying the radical expression. We can find the approximate distance by using a calculator to estimate when a radical expression is not a perfect square.

22. T HE DISTANCE AND MIDPOINT FORMULAS (11.3)

23. We can use the Distance formula to find the lengths of the sides of a geometric figure that is drawn on a geometric plane. Use the Distance formula to find the lengths of the sides then add them together. When adding values that are square roots, use a calculator to add before you round the answer. A (-2, 2) D (-3,- 2) C (3, -3) B (3,4) Sample Problem Find the exact length s of each side of the quadrilateral ABCD. Then find the perimeter to the nearest tenth.

24. T HE DISTANCE AND MIDPOINT FORMULAS (11.3)

25. If we know the coordinates of the endpoints of a diameter of a circle, we can use the midpoint formula to find the coordinates of the center of the circle. Sample Problem A circle is drawn on a coordinate plane. The endpoints of a diameter are (-1,5) and (4,-3). What is the center of the circle?

26. O PERATIONS WITH RADICAL EXPRESSION (11.4)

28. If both radical expressions have two terms, we can multiply the same way we find the product of two binomials, by using FOIL.

29. O PERATIONS WITH RADICAL EXPRESSION (11.4)

31. S OLVING RADICAL EQUATIONS (11.5) A radical equation is an equation that has a variable in a radicand. Solving a radical equation requires: 1 st : Get the radical by itself on one side of the equation. 2 nd : Square both sides. The radical expression under the radical must not be negative.

32. S OLVING RADICAL EQUATIONS (11.5)

37. G RAPHING SQUARE ROOT FUNCTIONS (11.6)

38. xy 00 11 21.4 42 62.4 93

39. G RAPHING SQUARE ROOT FUNCTIONS (11.6) For real numbers, the values of the radicand cannot be negative. So the domain is limited to those values of x that make the radicand greater than or equal to 0. To determine the domain values set the equation such that the radicand is greater than or equal to 0.

40. G RAPHING SQUARE ROOT FUNCTIONS (11.6)

42. Other changes to the square root function graph:

43. T RIGONOMETRIC R ATIOS (11.7) Trigonometric ratios : These are ratios of the lengths of the corresponding sides of similar right triangles and their relations to the corresponding angles. A C B c b a

44. T RIGONOMETRIC R ATIOS (11.7) Sample Problem Use the triangle below to find sin A, cos A, and tan A. b = 12 c = 13 a = 5

45. T RIGONOMETRIC R ATIOS (11.7) We can use the calculator to obtain values for the trig functions as well, given the angles. Make sure that your calculator is set on the “degree” mode. Sample Problem Find sin 50 o.

46. T RIGONOMETRIC R ATIOS (11.7) We can also use the trig. functions to find a missing length of a right triangle given the appropriate angles and lengths of sides. Sample Problem Find the value of x in the triangle at the right. 16 x 55 o

47. T RIGONOMETRIC R ATIOS (11.7) We can use the trigonometric ratios to measure distances indirectly when we know the angle of elevation or the angle of depression. Angle of elevation : An angle from the horizontal up to a line of sight. Angle of depression : An angle measured below the horizontal line of sight.

48. T RIGONOMETRIC R ATIOS (11.7) Sample Problem Suppose the angle of elevation from a rowboat to the light of a lighthouse is 35 o. You know that the lighthouse is 96 ft. tall. How far from the lighthouse is the rowboat.

49. T RIGONOMETRIC R ATIOS (11.7) Sample Problem A pilot is flying a plane 20,000 ft. above the ground. The pilot begins a 2 o descent to an airport runway. How far is the plane from the start of the runway (in ground distance)?

50. R ADICAL E XPRESSIONS AND EQUATIONS Chapter 11 THE END