1. Copyright © Ed2Net Learning, Inc.1 Good Afternoon! Today we will be learning about Review of Right Triangles Let’s warm up : Find the length of the missing sides in the given triangles. 45º 10 cm 1) 1) b = 10cm c = 14.14 cm 2) 2) b = 12.12 ft c = 14 ft 30º 60º 7 ft 45º 8 cm 3) 3) b = 8cm c = 11.31 cm 4) 4) b = 6.9 ft c = 8 ft 30º 60º 4 ft

2. Copyright © Ed2Net Learning, Inc. 2 In words: The square root of a number is one of its two equal factors. In Symbols: If x 2 = y, then x is a square root of y. Definition of square root Let’s review what we did in the last session. Square Roots and Irrational Numbers For example: The square root of 49 is 7 since 7.7 or 7 2 is 49. It is also true that -7.(-7) = 49. So -7 is another square root of 49.

3. Copyright © Ed2Net Learning, Inc. 3 The Square Root Symbol The symbol √, called the radical sign is used to indicate a non-negative square root. √49 indicates the non-negative square root of 49. √49 = 7 -√49 indicates the negative square root of 49. -√49 = -7

4. Copyright © Ed2Net Learning, Inc. 4 a.) √ 25 The symbol √ 25 represents the nonnegative square root of 25. Since 5*5 = 25, √ 25 = 5 Find each square root. b.) √ 100 The symbol √ 100 represents the nonnegative square root of 81. Since 10* 10 = 10, √ 100 = 10

5. Copyright © Ed2Net Learning, Inc. 5 The area of a square is 100 square inches. Find its perimeter. Area = 100 in 2 Answer : Perimeter is 40 inches First find the length of each side. A = s 2 100 = s 2 Replace A with 100. Both 10 and -10, when multiplied by themselves are 100. However, since length can not be negative, s must be 10. Hence, the length of each side is 10inches. Now find the perimeter. P = 4. s P = 4. 10 P = 40 Replace s with 10.

6. Copyright © Ed2Net Learning, Inc. 6 List some squares of numbers that are close to 125. 10 2 = 100 11 2 = 121 12 2 = 144 Numbers like 25, 49 and 64 are called perfect squares because when you take the square root then you get an answer that is whole number. What if the number is not perfect square? How do you find the square root of a number like 125? The number 125 is not a perfect square, that is, 125 has no square root hat is a whole number. Continued on the next slide.

7. Copyright © Ed2Net Learning, Inc. 7 Since 125 is close to 121 than to 144, the best whole number estimate for √125 would be 11. √ 121 < √125 < √144 √ 11 2 < √125 < √ 12 2 11 < √125 < 12 We know that 125 is greater than 121 or 11 2 and less than 144 or 12 2. So, the square root of 125 should be greater than 11 and less than 12.

8. Copyright © Ed2Net Learning, Inc. 8 Irrational Numbers Irrational Numbers: An irrational number is any real number that can not be expressed as a, b where a and b are integers and b is not equal to 0. Informally, this means numbers that cannot be represented as simple fractions. It can be deduced that they also cannot be represented as terminating or repeating decimals, Determine whether the number is rational or irrational. 0.22222... The three dots means that the 2s keep repeating. This is repeating decimal, so it can be expressed as a fraction. 0.22222... = 2 9 Thus, it is a rational number.

9. Copyright © Ed2Net Learning, Inc. 9 Rational number are numbers that can be expressed as a quotient as two integers, where the divisor are not zero. Irrational number are numbers that can be named by non terminating, non repeating decimals. Real numbers Rational numbers Integers Whole numbers Irrational numbers The sets of real numbers includes both the rational numbers and irrational numbers.

10. Copyright © Ed2Net Learning, Inc. 10 Solve the equation. Round decimal answer to the nearest tenth. y 2 = 75 Solution: y 2 = 75 y = √ 75 or - √75 Take the square root of each side. 75 = 8.660254038 y ≈ 8.7 or y ≈ -8.7

11. Copyright © Ed2Net Learning, Inc. 11 1) The area of a square is 144 square inches. Find its perimeter. 2) The area of a square is 256 square inches. Find its perimeter. Find the best integer estimate for each square root 3) √67 4) √45 1) 48 inches 2) 64 inches 3) 8 4) 7 Now you try!

12. Copyright © Ed2Net Learning, Inc. 12 Pythagorean Theorem The Pythagorean Theorem shows how the legs and hypotenuse of a right triangle are related. legs hypotenuse In a right triangle, the two shortest sides are legs. The longest side, which is opposite the right angle, is the hypotenuse.

13. Copyright © Ed2Net Learning, Inc. 13 In words: In a right angled triangle, the square of the length of the hypotenuse is equal to the sum of the square of the lengths of the legs. In Symbols: a 2 +b 2 = c 2. Pythagorean Theorem a b c If we know the lengths of two sides of a right angled triangle, then Pythagoras' Theorem allows us to find the length of the third side.

14. Copyright © Ed2Net Learning, Inc. 14 The lengths of the sides of a right triangle are 5 ft and 12 ft. Find the length of the hypotenuse. 1. Use the Pythagorean Theorem c 2 = a 2 +b 2 2. Replace a with 5 and b with 12 c 2 = 5 2 +12 2 = 25 +144 = 169 c= √ 169 = 13 The length of the hypotenuse = 13 ft.

15. Copyright © Ed2Net Learning, Inc. 15 You can use algebra to find any missing value, as in the following examples. a 2 + b 2 = c 2 9 2 + b 2 = 15 2 81 + b 2 = 225 Take 81 from both sides b 2 = 144 b = √144 b = 12 9 15 b

16. Copyright © Ed2Net Learning, Inc. 16 The converse of Pythagorean Theorem allows you to substitute the lengths of the sides of a triangle into the equation. c 2 = a 2 +b 2 To check whether a triangle is a right triangle, if the Pythagorean equation is true the triangle is a right triangle.

17. Copyright © Ed2Net Learning, Inc. 17 Is a triangle with sides 12 m, 15 m, and 20 m a right triangle? a 2 +b 2 = c 2 Write the equation for Pythagorean Theorem 12 2 +15 2 = 20 2 Replace a and b with the shorter lengths and c with the longest length 144+225 = 400 Simplify 369 ‡ 400 The triangle is not a right triangle.

18. Copyright © Ed2Net Learning, Inc. 18 The lengths of the sides of a right triangle are given. Find the length of the hypotenuse. 1) a = 16, b = 11 2) a = 23, b = 19 1) c = 19.4 2) c = 19.8 The measurement of three sides of a triangle are given. Determine whether each triangle is a right triangle. 3) 8 cm, 10 cm, 12.8cm 4) 7 ft, 9 ft, 13 ft 3) yes 4) no Now you try!

19. Copyright © Ed2Net Learning, Inc. 19 Distance and Midpoint Sometimes it is necessary to study line segments on the coordinate plane. A line segment, or a part of a line, contains two endpoints. The coordinates of these endpoints can help us find the length and the midpoint, or the point that is halfway between the two endpoints, of the line segment.

20. Copyright © Ed2Net Learning, Inc. 20 The Distance Formula To calculate the distance d of a line segment with endpoints (x 1, y 1 ) and (x 2, y 2 ) use the formula d = (x 2 - x 1 ) 2 + (y 2 - y 1 ) 2. We can calculate the length of a line segment by using the Distance Formula.

21. Copyright © Ed2Net Learning, Inc. 21 Example Find the distance between (2, 3) and (6, 8). Let x 1 = 2, x 2 = 6, y 1 = 3, and y 2 = 8. d = (x 2 - x 1 ) 2 + (y 2 - y 1 ) 2. d = (6 - 2) 2 + (8 - 3) 2. d = (4) 2 + (5) 2. d = 16 + 25. d = 41. = 6.4 units The Distance Formula Substitute. Add Calculate square root.

22. Copyright © Ed2Net Learning, Inc. 22 Find the distance between the points (1, -7) and (-5, -3). The distance formula for the distance between two points P (x 1, y 1 ) and Q (x 2, y 2 ) is as the following: PQ = (x 2 - x 1 ) 2 + (y 2 - y 1 ) 2. PQ = (-5 - 1) 2 + (-3 – (-7)) 2. PQ = (-6) 2 + (4) 2 = 36 +16 = 52 = 7.21 units

23. Copyright © Ed2Net Learning, Inc. 23 The Midpoint Formula To calculate the midpoint of a line segment with endpoints (x 1, y 1 ) and (x 2, y 2 ) use the formula (x 1 + x 2 ), (y 1 + y 2 ) 2 2 We can calculate the midpoint of a line segment by using the Midpoint Formula.

24. Copyright © Ed2Net Learning, Inc. 24 Find the midpoint of (5, 1) and (-1, 5). Let x 1 = 5, x 2 = -1, y 1 = 1, and y 1 = 5. (x 1 + x 2 ), (y 1 + y 2 ) 2 2 The Midpoint Formula = (5 + -1), (1 + 5 ) 2 2 Substitute. Add = (4), (6 ) 2 2 (2, 3) is the midpoint. = (2, 3)

25. Copyright © Ed2Net Learning, Inc. 25 Find the midpoint between the points (1, -7) and (-5, -3). The midpoint formula for the coordinates of the midpoint M (x, y) between two points P (x 1, y 1 ) and Q (x 2, y 2 ) is as the following: M (x, y) = (x 1 + x 2 ), (y 1 + y 2 ) 2 2 M (x, y) = M {(1 + (-5)), ((-7) + (-3) )} 2 2 M (x, y) = M {(-4), (-10)} = M(-2, -5) 2 2 (-2, -5) is the midpoint.

26. Copyright © Ed2Net Learning, Inc. 26 0 A(4, 2) B(-2, 2) C(0, -1) y x Find the perimeter of ΔABC. We can calculate the length of a line segment by using the Distance Formula. AB = (4 – (-2)) 2 + (2 - 2) 2. AB = (6) 2 + (0) 2 = 6 units.

27. Copyright © Ed2Net Learning, Inc. 27 BC = (0 - (-2)) 2 + ((-1) - 2) 2. BC = (2) 2 + (-3) 2 BC = 4 + 9 = 13 = 3.6 units. CA = (4 - 0)) 2 + (2 – (-1)) 2. CA = (4) 2 + (3) 2 CA = 16 + 9 = 25 = 5 units. The perimeter of ΔABC = length AB + length BC + length CA = 6 + 3.6 + 5 = 14.6 The perimeter of ΔABC = 14.6 units.

28. Copyright © Ed2Net Learning, Inc. 28 Find the distance between each pair of points. Round answers to the nearest hundredth. 1) (5, 3), (9, 9) 2) (14, 6), (12, 7) 1) 7.2 units2) 2.23 units Find the midpoint of the given points. 3) (5, 3), (9, 9) 4) (14, 6), (12, 7) 3) (7, 6)4) (13, 6.5) Now you try!

29. Copyright © Ed2Net Learning, Inc. 29 BREAK

30. Copyright © Ed2Net Learning, Inc. 30 GAME Click on the link below for some exciting puzzle http://www.thekidzpage.com/onlinejigsawpuzzles/jigsaw -puzzles/12-piece-jigsaw/121505bows.html

31. Copyright © Ed2Net Learning, Inc. 31 Multiplying square roots For nonnegative numbers, the square root of a product equals the product of the square roots. If a ≥ b and b ≥ 0, then √ab = √a. √b The rule for multiplying square roots is specially useful with an isosceles right triangle, which is also known by its angle measures as a 45º-45º-90º triangle. If you draw an diagonal in a square, you will form two congruent 45º-45º-90º triangles. 45º Special Right Triangles

32. Copyright © Ed2Net Learning, Inc. 32 You can use the rule to relate the lengths of the sides and the hypotenuse in such a triangle. Use the Pythagorean theorem c 2 = a 2 + b 2 c 2 = x 2 + x 2 Replace a and b with x c 2 = 2. x 2 Simplify c = 2. x 2 c = √2. √ x 2 Use the rule for multiplying square roots c = √2. x or x. √2 Simplify 45º-45º-90º triangle

33. Copyright © Ed2Net Learning, Inc. 33 In a 45º-45º-90º triangle, the legs are congruent and the length of the hypotenuse is the length of the leg times √2. hypotenuse = leg. √2 This shows the following relationship. 45º 1 1 √2 45º-45º-90º triangle

34. Copyright © Ed2Net Learning, Inc. 34 Find the length of AC in ΔABC. 45º 5cm A B C c = a √2 c = 5 √2 Replace a with 5 c = 5 x 1.414 c = 7.07 cm 5cm The length of AC in ΔABC is 7.07 cm.

35. Copyright © Ed2Net Learning, Inc. 35 30º-60º-90º triangle Another special triangle is the 30º-60º-90º triangle. You can form two congruent 30º-60º-90º triangles by bisecting an angle of an equilateral triangle. This is shown in the diagram. 60º 30º A B C D xx 2x In the diagram, the length of the hypotenuse of each 30º-60º-90º triangle is twice the length of the shorter leg. You can use the Pythagorean theorem to find the length of the longer length.

36. Copyright © Ed2Net Learning, Inc. 36 For the figure at the right, find the length of the longer side. 30º 60º b A B C x 2x (2x) 2 = x 2 + b 2 Use the Pythagorean theorem 4x 2 = x 2 + b 2 Simplify 3x 2 = b 2 Subtract x 2 from each side √3x 2 = b Find the square root √3. x = b Rule for multiplying square roots b = √3. x or x. √3 Simplify 30º-60º-90º triangle

37. Copyright © Ed2Net Learning, Inc. 37 This shows the special relationship of hypotenuse and the legs in a 30º-60º-90º triangle. 30º-60º-90º triangle In a 30º-60º-90º triangle, the length of the hypotenuse is the 2 times length of the shorter leg. The length of the longer leg is the length of the shorter leg times √3. 30º 60º 2s A B C s s √3 hypotenuse = 2. shorter leg longer leg = shorter leg. √2

38. Copyright © Ed2Net Learning, Inc. 38 Find the missing length in ΔABC. 30º 60º y A B C 5ft x Hypotenuse = 2. shorter length x= 2. 5 The length of the shorter le is 5. x= 10 Simplify Longer leg = shorter leg. √3 y= 5. √3 Replace a with 5 y ≈ 8.7 The length of hypotenuse is 10 ft, and he length of the longer leg is about 8.7 ft.

39. Copyright © Ed2Net Learning, Inc. 39 Now you try! Find the length of the missing sides in the given triangles. 45º 13 cm 1) 1) b = 13cm c = 18.4 cm 2) 2) b = 22.5 ft c = 26 ft 30º 60º 13 ft 45º 7 cm 3) 3) b = 8cm c = 9.9 cm 4) 4) b = 36.3 ft c = 42 ft 30º 60º 21 ft

40. Copyright © Ed2Net Learning, Inc. 40 You have done a nice job. See you in the next session.