Slide 1 / 145 Slide 2 / 145 8th Grade Pythagorean Theorem, Distance & Midpoint 2016-01-15 www.njctl.org Table of Contents Slide 3 / 145 Proofs Click on a topic to go to that section Pythagorean Theorem Distance Formula Midpoints Glossary & Standards
Slide 4 / 145 Proofs Click to return to the table of contents Introduction to Proofs Slide 5 / 145 Lab: Introduction to Proofs 1 Determine which type of sandwich you would like to start. Choose only one. Slide 6 / 145 A Peanut Butter & Jelly B Ham and Cheese C Fluffernutter D Tuna melt BLT E Chicken Salad BLT F Peanut Butter & Banana G Bologna and Cheese H Grilled Cheese
2) Fill in the blank with the winner of the class poll. Everyday for lunch, you make yourself a sandwich. The sandwich that you are making today is Slide 7 / 145. 3) What steps do you need to take in order to make your sandwich? List all of the thoughts of your group in the space provided on the Lab WS. 4) List any additional thoughts provided by the rest of the groups in the class. Slide 8 / 145 5) List all of the thoughts contributed by your group and the rest of the class in the chronological order decided upon by the class. Slide 9 / 145
6) Was the sandwich made correctly? Why or why not? If not, what steps do you need to add? Slide 10 / 145 7) Write down the revised list all of the thoughts contributed by your group and the rest of the class in the chronological order. Slide 11 / 145 Proofs Slide 12 / 145 A proof involves writing reasoned, logical explanations that use definitions, algebraic properties, postulates, and previously proven theorems to arrive at a conclusion. A postulate is a property that is accepted without proof, usually because one can easily understand why it is true. One of the classic postulates of Geometry states that "Through any two points, there is exactly one line". A B A B
Proofs Slide 13 / 145 The lab that you completed about making a sandwich has the same general steps as a proof. You brainstorm and write down various ideas based on what you know, and organize them into the proper order that will lead you to the final conclusion. You can also use proofs when solving algebraic equations. As you write each step/idea, you also write down the definition, property, postulate, etc. that justifies your step/idea. The example shown below is written as a two-column proof. Given: 3x + 2 = 14 Prove: x = 4 Statements 1. 3x + 2 = 14 1. Given Reasons 2. 3x = 12 2. Subtraction 3. x = 4 3. Division Note: The first step to solving a proof is always "Given" to you. Proofs Slide 14 / 145 When writing two-column proofs, you can also combine the rules of algebra and properties of geometry. B Example: Given: m A = (4x + 3), m B = (5x + 4), (5x + 4) m C = (x + 23) Prove: ΔABC is an acute triangle (4x + 3) (x + 23) A C Proofs B Slide 15 / 145 Given: m A = (4x + 3), m B = (5x + 4), (5x + 4) m C = (x + 23) Prove: ΔABC is an acute triangle (4x + 3) (x + 23) A C Statements Reasons 1. m A = (4x + 3), m B = (5x + 4), m C = (x + 23) 1. Given 2. m A + m B + m C = 180 2. All of the interior angles of a triangle sum to 180 3. 4x + 3 + 5x + 4 + x + 23 = 180 3. Substitution 4. 10x + 30 = 180 4. Combine Like Terms 5. 10x = 150 5. Subtraction 6. x = 15 6. Division 7. m A = 4(15) + 3 = 63, m B = 5(15) + 4 = 79 7. Substitution m C = (15) + 23 = 38 8. ΔABC is an acute triangle 8. Definition of an acute Δ
Proofs Slide 16 / 145 Example: Write a two-column proof. Given: 2(x + 4) + 4(x - 5) = -42 Prove: x = -5 Statements Reasons Proofs E Example: Given: m D = (4x + 7), m E = (9x + 5), (9x + 5) m F = (2x + 3) Prove: ΔDEF is an obtuse Δ (4x + 7) (2x + 3) D Statements Reasons F Slide 17 / 145 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7 Proofs The next 10 Response Questions are all based on the problem below. If needed, fill in the two-column proof as you answer each question. Slide 18 / 145 Statements Reasons
2 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7 Slide 19 / 145 What is the first statement? A -6x - 18 + 3x + 15 = -18 B -6x + 18 + 3x - 15 = -18 C -6(x - 3) + 3(x - 5) = -18 D -6x - 18 + 3x - 15 = -18 3 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7 Slide 20 / 145 What is the first reason? A Distributive Property B Addition C Multiplication D Given 4 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7 Slide 21 / 145 What is the second statement? A -6x - 18 + 3x + 15 = -18 B -6x + 18 + 3x - 15 = -18 C -6(x - 3) + 3(x - 5) = -18 D -6x - 18 + 3x - 15 = -18
5 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7 Slide 22 / 145 What is the second reason? A Distributive Property B Addition C Multiplication D Given 6 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7 Slide 23 / 145 What is the third statement? A -3x - 3 = -18 B 9x - 3 = -18 C 9x + 3 = -18 D -3x + 3 = -18 7 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7 Slide 24 / 145 What is the third reason? A Simplify or Combine Like Terms B Addition C Division D Subtraction
8 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7 Slide 25 / 145 What is the fourth statement? A -3x = -15 B -3x = -21 C 9x = -15 D 9x = -21 9 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7 Slide 26 / 145 What is the fourth reason? A Simplify or Combine Like Terms B Addition C Division D Subtraction 10 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7 Slide 27 / 145 What is the fifth statement? A x = 7 B x = 5 C x = - D x = - 7 3 5 3
11 Given: -6(x - 3) + 3(x - 5) = -18. Prove: x = 7 Slide 28 / 145 What is the fifth reason? A Simplify or Combine Like Terms B Addition C Division D Subtraction Proofs The next 8 Response Questions are all based on the problem below. If needed, fill in the two-column proof as you answer each question. C (6x + 7) (11x + 3) A B D E Slide 29 / 145 Given: ABC and CBD are supplementary Prove: ABC EFG. Statements 1. ABC and CBD are supplementary 2. m ABC + m CBD = 180 3. 6x + 7 + 11x + 3 = 180 4. 17x + 10 = 180 5. 17x = 170 6. x = 10 7. m ABC = 6(10) + 7 = 67, m EFG = 7(10) - 3 = 67 8. ABC m EFG F (7x - 3) G Reasons 1. 2. 3. 4. 5. 6. 7. 8. 12 Given: ABC and CBD are supplementary Prove: ABC EFG What is the first reason? A Definition of supplementary angles Slide 30 / 145 B Addition C Substitution D Division E Definition of congruent angles F Given G Subtraction H Combine Like Terms
13 Given: ABC and CBD are supplementary Prove: ABC EFG What is the second reason? A Definition of supplementary angles Slide 31 / 145 B Addition C Substitution D Division E Definition of congruent angles F Given G Subtraction H Combine Like Terms 14 Given: ABC and CBD are supplementary Prove: ABC EFG What is the third reason? A Definition of supplementary angles Slide 32 / 145 B Addition C Substitution D Division E Definition of congruent angles F Given G Subtraction H Combine Like Terms 15 Given: ABC and CBD are supplementary Prove: ABC EFG What is the fourth reason? A Definition of supplementary angles Slide 33 / 145 B Addition C Substitution D Division E Definition of congruent angles F Given G Subtraction H Combine Like Terms
16 Given: ABC and CBD are supplementary Prove: ABC EFG What is the fifth reason? A Definition of supplementary angles Slide 34 / 145 B Addition C Substitution D Division E Definition of congruent angles F Given G Subtraction H Combine Like Terms 17 Given: ABC and CBD are supplementary Prove: ABC EFG What is the sixth reason? A Definition of supplementary angles Slide 35 / 145 B Addition C Substitution D Division E Definition of congruent angles F Given G Subtraction H Combine Like Terms 18 Given: ABC and CBD are supplementary Prove: ABC EFG What is the seventh reason? A Definition of supplementary angles Slide 36 / 145 B Addition C Substitution D Division E Definition of congruent angles F Given G Subtraction H Combine Like Terms
19 Given: ABC and CBD are supplementary Prove: ABC EFG What is the seventh reason? A Definition of supplementary angles Slide 37 / 145 B Addition C Substitution D Division E Definition of congruent angles F Given G Subtraction H Combine Like Terms Slide 38 / 145 Pythagorean Theorem Click to return to the table of contents Pythagorean Theorem Slide 39 / 145 Pythagorean theorem is used for right triangles. It was first known in ancient Babylon and Egypt beginning about 1900 B.C. However, it was not widely known until Pythagoras stated it. Pythagoras lived during the 6th century B.C. on the island of Samos in the Aegean Sea. He also lived in Egypt, Babylon, and southern Italy. He was a philosopher and a teacher.
a Labels for a right triangle c Hypotenuse click to reveal - Opposite the right angle - Longest click of the to 3 reveal sides Slide 40 / 145 Legs click to reveal b - 2 sides that form the right angle click to reveal Pythagorean Theorem Proofs Slide 41 / 145 In a right triangle, the sum of the squares of the lengths of the legs (a and b) is equal to the square of the length of the hypotenuse (c). a 2 + b 2 = c 2 Click on the links below to see several animations of the proof Water demo Move slider to show c 2 Moving of squares Proof of the Pythagorean Theorem Slide 42 / 145 Lab: Proof of the Pythagorean Theorem
Follow-up Questions: Slide 43 / 145 11. How is the area of the printed square on page 1 related to the area of the printed square on page 2? Explain how you reached that conclusion. 12. How are the 4 right triangles that you cut out at the beginning of this lab related to each other? Explain how you reached that conclusion. Follow-up Questions (cont'd): Slide 44 / 145 13. How are the areas that you found in question #6 & question #10 related to each other? Explain how you reached that conclusion. 14. What algebraic expression represents the side length of the printed squares. Explain how you reached that conclusion. Follow-up Questions (cont'd): 15. Multiply these side lengths together and simplify the expression. Slide 45 / 145 16. Using the arrangement of the shapes from question #5, write an algebraic expression to represent the area of the entire figure.
Follow-up Questions (cont'd): Slide 46 / 145 17. Set the expressions from question #15 & question #16 equal to one another and simplify the equation. Pythagorean Theorem Slide 47 / 145 How to use the formula to find missing sides. Missing Leg Write Equation Substitute in numbers Square numbers Subtract Find the Square Root Label Answer Missing Hypotenuse Write Equation Substitute in numbers Square numbers Add Find the Square Root Label Answer Pythagorean Theorem Slide 48 / 145 Missing Leg 5 ft 15 ft a 2 + b 2 = c 2 5 2 + b 2 = 15 2 25 + b 2 = 225-25 -25 b 2 = 200 Write Equation Substitute in numbers Square numbers Subtract Find the Square Root Label Answer
Pythagorean Theorem Slide 49 / 145 Missing Leg a 2 + b 2 = c 2 Write Equation 9 in 18 in 9 2 + b 2 = 18 2 Substitute in numbers 81 + b 2 = 324 Square numbers -81-81 Subtract b 2 = 243 Find the Square Root Label Answer Pythagorean Theorem Slide 50 / 145 Missing Hypotenuse a 2 + b 2 = c 2 Write Equation 7 in 4 2 + 7 2 = c 2 Substitute in numbers 2 16 + 49 = c Square numbers 4 in 65 = c 2 Add Find the Square Root & Label Answer 20 What is the length of the third side? Slide 51 / 145 7 x 4
21 What is the length of the third side? Slide 52 / 145 x 41 15 22 What is the length of the third side? Slide 53 / 145 7 x 4 23 What is the length of the third side? Slide 54 / 145 3 x 4
Pythagorean Triples Slide 55 / 145 3 5 There are combinations of whole numbers that work in the Pythagorean Theorem. These sets of numbers are known as Pythagorean Triples. 4 3-4-5 is the most famous of the triples. If you recognize the sides of the triangle as being a triple (or multiple of one), you won't need a calculator! Pythagorean Triples Slide 56 / 145 Can you find any other Pythagorean Triples? Use the list of squares to see if any other triples work. 1 2 = 1 11 2 = 121 21 2 = 441 2 2 = 4 12 2 = 144 22 2 = 484 3 2 = 9 13 2 = 169 23 2 = 529 4 2 = 16 14 2 = 196 24 2 = 576 5 2 = 25 15 2 = 225 25 2 = 625 6 2 = 36 16 2 = 256 26 2 = 676 7 2 = 49 17 2 = 289 27 2 = 729 8 2 = 64 18 2 = 324 28 2 = 784 9 2 = 81 19 2 = 361 29 2 = 841 10 2 = 100 20 2 = 400 30 2 = 900 Triples 24 What is the length of the third side? Slide 57 / 145 6 8
25 What is the length of the third side? Slide 58 / 145 5 13 26 What is the length of the third side? Slide 59 / 145 48 50 27 The legs of a right triangle are 7.0 and 3.0, what is the length of the hypotenuse? Slide 60 / 145
28 The legs of a right triangle are 2.0 and 12, what is the length of the hypotenuse? Slide 61 / 145 29 The hypotenuse of a right triangle has a length of 4.0 and one of its legs has a length of 2.5. What is the length of the other leg? Slide 62 / 145 30 The hypotenuse of a right triangle has a length of 9.0 and one of its legs has a length of 4.5. What is the length of the other leg? Slide 63 / 145
This is a great problem and draws on a lot of what we've learned. Slide 64 / 145 Try it in your groups. Then we'll work on it step by step together by asking questions that break the problem into pieces. In ΔABC, BD is perpendicular to AC. The dimensions are shown in centimeters. B 10 10 8 A D C What is the length of AC? From PARCC EOY sample test calculator #1 31 What have we learned that will help solve this problem? A Pythagorean Theorem Slide 65 / 145 B Pythagorean Triples C Distance Formula D A and B only In ΔABC, BD is perpendicular to AC. The dimensions are shown in centimeters. B 10 10 8 A D C What is the length of AC? First, notice that we have two right triangles (perpendicular lines make right angles). The triangles are outlined red & blue in the diagram below. Slide 66 / 145 In ΔABC, BD is perpendicular to AC. The dimensions are shown in centimeters. B 10 10 8 A D C What is the length of AC?
32 What is the length of the 3rd side in the red triangle? A 3 cm Slide 67 / 145 B 6 cm C 9 cm D 13.45 cm In ΔABC, BD is perpendicular to AC. The dimensions are shown in centimeters. B 10 10 8 A D C What is the length of AC? 33 How is AD related to CD? A AD > CD Slide 68 / 145 B AD < CD C AD = CD D not enough information to relate these segments In ΔABC, BD is perpendicular to AC. The dimensions are shown in centimeters. B 10 10 8 A 6 D C What is the length of AC? 34 What is the length of AC? Slide 69 / 145 In ΔABC, BD is perpendicular to AC. The dimensions are shown in centimeters. B 10 10 8 A D C What is the length of AC?
Converse of the Pythagorean Theorem Slide 70 / 145 If a and b are measures of the shorter sides of a triangle, c is the measure of the longest side, and c 2 = a 2 + b 2, then the triangle is a right triangle. If c 2 a 2 + b 2, then the triangle is not a right triangle. This is the Converse of the Pythagorean Theorem. a = 3 ft c = 5 ft b = 4 ft Converse of the Pythagorean Theorem Slide 71 / 145 In other words, you can check to see if a triangle is a right triangle by seeing if the Pythagorean Theorem is true. Test the Pythagorean Theorem. If the final equation is true, then the triangle is right. If the final equation is false, then the triangle is not right. Converse of the Pythagorean Theorem Slide 72 / 145 8 in, 17 in, 15 in a 2 + b 2 = c 2 8 2 + 15 2 = 17 2 64 + 225 = 289 289 = 289 Yes! Is it a Right Triangle? Write Equation Plug in numbers Square numbers Simplify both sides Are they equal?
35 Is the triangle a right triangle? Slide 73 / 145 Yes No 6 ft 10 ft 8 ft 36 Is the triangle a right triangle? Slide 74 / 145 Yes No 24 ft 36 ft 30 ft 37 Is the triangle a right triangle? Slide 75 / 145 Yes No 8 in. 12 in. 10 in.
38 Is the triangle a right triangle? Slide 76 / 145 Yes No 5 ft 13 ft 12 ft 39 Can you construct a right triangle with three lengths of wood that measure 7.5 in, 18 in and 19.5 in? Slide 77 / 145 Yes No Applications of Pythagorean Theorem Slide 78 / 145 Steps to Pythagorean Theorem Application Problems. 1. Draw a right triangle to represent the situation. 2. Solve for unknown side length. 3. Round to the nearest tenth.
Applications of Pythagorean Theorem Slide 79 / 145 Work with your partners to complete: To get from his high school to his home, Jamal travels 5.0 miles east and then 4.0 miles north. When Sheila goes to her home from the same high school, she travels 8.0 miles east and 2.0 miles south. What is the measure of the shortest distance, to the nearest tenth of a mile, between Jamal's home and Sheila's home? -10-5 y 10 5 0-5 -10 5 10 x From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011. Applications of Pythagorean Theorem Slide 80 / 145 Work with your partners to complete: A straw is placed into a rectangular box that is 3 inches by 4 inches by 8 inches, as shown in the accompanying diagram. If the straw fits exactly into the box diagonally from the bottom left front corner to the top right back corner, how long is the straw, to the nearest tenth of an inch? From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011. Applications of Pythagorean Theorem Slide 81 / 145 The Pythagorean Theorem can be applied to 3 Dimensional Figures In this figure: a = slant height (height of triangular face) b = 1/2 base length (from midpoint of side of base to center of the base of the pyramid) h = height of pyramid
Applications of Pythagorean Theorem Slide 82 / 145 A right triangle is formed between the three lengths. If you know two of the measurements, you can calculate the third. EXAMPLE: Find the slant height of a pyramid whose height is 5 cm and whose base has a length of 8cm. Applications of Pythagorean Theorem Slide 83 / 145 Find the slant height of the pyramid whose base length is 10 cm and height is 12 cm. Label the diagram with the measurements. Applications of Pythagorean Theorem Slide 84 / 145 Find the base length of the pyramid whose height is 21 m and slant height is 29 m. Label the diagram with the measurements.
40 The sizes of television and computer monitors are given in inches. However, these dimensions are actually the diagonal measure of the rectangular screens. Suppose a 14-inch computer monitor has an actual screen length of 11-inches. What is the height of the screen? Slide 85 / 145 41 Find the height of the pyramid whose base length is 16 in and slant height is 17 in. Label the diagram with the measurements. Slide 86 / 145 42 A tree was hit by lightning during a storm. The part of the tree still standing is 3 meters tall. The top of the tree is now resting 8 meters from the base of the tree, and is still partially attached to its trunk. Assume the ground is level. How tall was the tree originally? Slide 87 / 145
43 Suppose you have a ladder of length 13 feet. To make it sturdy enough to climb you myct place the ladder exactly 5 feet from the wall of a building. You need to post a banner on the building 10 feet above ground. Is the ladder long enough for you to reach the location you need to post the banner? Yes Slide 88 / 145 No (Derived from ( 44 You've just picked up a ground ball at 3rd base, and you see the other team's player running towards 1st base. How far do you have to throw the ball to get it from third base to first base, and throw the runner out? (A baseball diamond is a square) Slide 89 / 145 2nd 90 ft. 90 ft. 3rd 1st 90 ft. 90 ft. home 45 You're locked out of your house and the only open window is on the second floor, 25 feet above ground. There are bushes along the edge of your house, so you'll have to place a ladder 10 feet from the house. What length of ladder do you need to reach the window? Slide 90 / 145
46 Scott wants to swim across a river that is 400 meters wide. He begins swimming perpendicular to the shore, but ends up 100 meters down the river because of the current. How far did he actually swim from his starting point? Slide 91 / 145 Slide 92 / 145 Distance Formula Click to return to the table of contents Distance Between Two Points If you have two points on a graph, such as (5,2) and (5,6), you can find the distance between them by simply counting units on the graph, since they lie in a vertical line. y Slide 93 / 145 10 The distance between these two points is 4. 5 The top point is 4 above the lower point. -10-5 0 5 10 x -5-10
47 What is the distance between these two points? y Slide 94 / 145 10 5-10 -5 0 5 10 x -5-10 48 What is the distance between these two points? y Slide 95 / 145 10 5-10 -5 0 5 10 x -5-10 49 What is the distance between these two points? y Slide 96 / 145 10 5-10 -5 0 5 10 x -5-10
Distance Between Two Points Most sets of points do not lie in a vertical or horizontal line. For example: y Slide 97 / 145-10 -5 10 5 0-5 5 10 x Counting the units between these two points is impossible. So mathematicians have developed a formula using the Pythagorean theorem to find the distance between two points. -10 Distance Between Two Points Draw the right triangle around these two points. Then use the Pythagorean theorem to find the distance in red. y Slide 98 / 145 10 5 c b -10-5 0 a 5 10 x -5-10 Distance Between Two Points Slide 99 / 145 Example: y 10 5-10 -5 0 5 10 x -5-10
Distance Between Two Points Slide 100 / 145 Try This: y 10 5-10 -5 0 5 10 x -5-10 Distance Formula Slide 101 / 145 Deriving a formula for calculating distance... Distance Formula Slide 102 / 145 Create a right triangle around the two points. Label the points as shown. Then substitute into the Pythagorean Theorem. y -10-5 10 (x 2, y 2) 5 d length = y 2 - y 1 (x 1, y 1) (x 2, y 1) x 0 10 length = x 2 - x 1-5 c 2 = a 2 + b 2 d 2 = (x 2 - x 1) 2 + (y 2 - y 1) 2 d = (x 2 - x 1) 2 + (y 2 - y 1) 2 This is the distance formula now substitute in values. -10
Distance Formula Slide 103 / 145 You can find the distance d between any two points (x 1, y 1) and (x 2, y 2) using the formula below. d = (x 2 - x 1 ) 2 + (y 2 - y 1 ) 2 how far between the x-coordinate how far between the y-coordinate Distance Formula Slide 104 / 145 When only given the two points, use the formula. Find the distance between: Point 1 (-4, -7) Point 2 (-5, -2) 50 Find the distance between (2, 3) and (6, 8). Round answer to the nearest tenth. Slide 105 / 145 hint Let: x 1 = 2 y 1 = 3 x 2 = 6 y 2 = 8
51 Find the distance between (-7, -2) and (11, 3). Round answer to the nearest tenth. Slide 106 / 145 hint Let: x 1 = -7 y 1 = -2 x 2 = 11 y 2 = 3 52 Find the distance between (4, 6) and (1, 5). Round answer to the nearest tenth. Slide 107 / 145 53 Find the distance between (7, -5) and (9, -1). Round answer to the nearest tenth. Slide 108 / 145
Applications of the Distance Formula How would you find the perimeter of this rectangle? Slide 109 / 145 Either just count the units or find the distance between the points from the ordered pairs. Applications of the Distance Formula Slide 110 / 145 Can we just count how many units long each line segment is in this quadrilateral to find the perimeter? D (3,3) C (9,4) A (0,-1) B (8,0) Applications of the Distance Formula Slide 111 / 145 You can use the Distance Formula to solve geometry problems. D (3,3) A (0,-1) C (9,4) B (8,0) Find the perimeter of ABCD. Use the distance formula to find all four of the side lengths. Then add then together. AB = AB = BC = BC = CD = CD = DA = DA =
54 Find the perimeter of ΔEFG. Round the answer to the nearest tenth. Slide 112 / 145 F (3,4) G (1,1) E (7,-1) 55 Find the perimeter of the square. Round answer to the nearest tenth. Slide 113 / 145 H (1,5) K (-1,3) I (3,3) J (1,1) 56 Find the perimeter of the parallelogram. Round answer to the nearest tenth. Slide 114 / 145 L (1,2) M (6,2) O (0,-1) N (5,-1)
Slide 115 / 145 Midpoints Click to return to the table of contents Midpoint Slide 116 / 145 10 y (2, 10) Find the midpoint of the line segment. 05 5 0 (2, 2) 5 10 x What is a midpoint? How did you find the midpoint? What are the coordinates of the midpoint? -5-10 Midpoint Slide 117 / 145 y Find the midpoint of the line segment. 10 What are the coordinates of the midpoint? 5 (3, 4) (9, 4) How is it related to the coordinates of the endpoints? 0 5 10 x -5-10
The Midpoint Formula Slide 118 / 145 To calculate the midpoint of a line segment with endpoints (x 1,y 1) and (x 2,y 2) use the formula: ( x1 + x 2 y 1 + y 2, 2 2 ) The x and y coordinates of the midpoint are the averages of the x and y coordinates of the endpoints, respectively. The Midpoint Formula Slide 119 / 145 The midpoint of a segment AB is the point M on AB halfway between the endpoints A and B. A (2,5) B (8,1) See next page for answer The Midpoint Formula Slide 120 / 145 The midpoint of a segment AB is the point M on AB halfway between the endpoints A and B. A (2,5) M B (8,1) Use the midpoint formula: + x 2 y 1 + y 2, 2 2 ) ( x1
The Midpoint Formula Slide 121 / 145 Find the midpoint of (1, 0) and (-5, 3). Use the midpoint formula: + x 2 y 1 + y 2, 2 2 ) ( x1 57 What is the midpoint of the line segment that has the endpoints (2, 10) and (6,-4)? y A (3, 4) 10 B (4, 7) C (4, 3) 5 D (1.5, 3) -10-5 0 5 10-5 x Slide 122 / 145-10 58 What is the midpoint of the line segment that has the endpoints (4, 5) and (-2, 6)? y A (3, 6.5) 10 B (1, 5.5) C (-1, 5.5) 5 D (1, 0.5) x -10-5 0 5 10 Slide 123 / 145-5 -10
59 What is the midpoint of the line segment that has the endpoints (-4, -7) and (-12, 2)? Slide 124 / 145 A (-8, -2.5) B (-4, -4.5) C (-1, -6.5) D (-8, -4) 60 What is the midpoint of the line segment that has the endpoints (10, 9) and (5, 3)? Slide 125 / 145 A (6.5, 2) B (6, 7.5) C (7.5, 6) D (15, 12) 61 Find the center of the circle with a diameter having endpoints at (-4, 3) and (0, 2). Slide 126 / 145 Which formula should be used to solve this problem? A Pythagorean Formula B Distance Formula C D Midpoint Formula Formula for Area of a Circle
62 Find the center of the circle with a diameter having endpoints at (-4, 3) and (0, 2). A (2.5,-2) B (2,2.5) C (-2,2.5) D (-1,1.5) Slide 127 / 145 63 Find the center of the circle with a diameter having endpoints at (-12, 10) and (2, 6). A (-7,8) B (-5,8) C (5,8) D (7,8) Slide 128 / 145 Using Midpoint to Find the Missing Endpoint If point M is the midpoint between the points P and Q. Find the coordinates of the missing point. Q =? Slide 129 / 145 M (8, 1) P (8, -6) Use the midpoint formula and solve for the unknown. + x 2 y 1 + y 2, 2 2 ) ( x1 Substitute Multiply both sides by 2 Add or subtract (8, 8)
Using Midpoint to Find the Missing Endpoint Slide 130 / 145 If point M is the midpoint between the points P and Q. Find the coordinates of the missing point. Another method that can be used to find the missing endpoint is to look at the relationship between both the x- and y-coordinates and use the relationship again to calculate the missing endpoint. Q =? +0 +7 M (8, 1) +0 +7 P (8, -6) Following the pattern, we see that the coordinates for point Q are (8, 8), which is exactly the same answer that we found using the midpoint formula. 64 If Point M is the midpoint between the points P and Q. What are the coordinates of the missing point? Slide 131 / 145 A (-13, -22) B (-8.5, -9.5) C (-4.5, -7.5) D (-12.5, -6.5) P = (-4,3) M = (-8.5,-9.5) Q =? 65 If Point M is the midpoint between the points P and Q. What are the coordinates of the missing point? Slide 132 / 145 A (1, -1) B (-13, 19) C (-8, 11) D (-19, 8) Q = (-6, 9) M = (-7, 10) P =?
Slide 133 / 145 Glossary & Standards Click to return to the table of contents Converse of Pythagorean Theorem If a and b are measures of the shorter sides of a triangle, c is the measure of the longest side, and c squared equals a squared plus b squared, then the triangle is a right triangle. a a 2 +b 2 = c 2 c 4 3 5 Example: 4 2 +3 2 = 5 2 16+9 = 25 25 = 25 Slide 134 / 145 b right triangle Back to Instruction Distance Length Measurement of how far two points are through space. Slide 135 / 145 Distance Formula: 2 2 10 2 2 0 2 10 8 8 642 0= 28 10 8 8 Back to Instruction
Hypotenuse Slide 136 / 145 The longest side of a right triangle that is opposite the right angle. a 2 +b 2 = c 2 Back to Instruction Leg Slide 137 / 145 2 sides that form the right angle of a right triangle. a 2 +b 2 = c 2 Back to Instruction Midpoint Slide 138 / 145 The middle of something. The point halfway along a line. ( x1 Midpoint Formula: + x 2 y 1 + y 2, 2 2 ) ( x1 + x2 y1 + y2, 2 2 ) ( 2 + 2 10 + 2, 2 2 ) ( 4, 12 2 2 ) ( 6 2, ) Back to Instruction
Postulate Slide 139 / 145 A property that is accepted without proof Through any two points, there is exactly one line. Segment Addition Postulate E Angle Addition Postulate J F A B C D CD + DE = CE G H m FGJ + m JGH = m FGH Back to Instruction Proof Reasoned, logical explanations that use definitions, algebraic properties, postulates, and previously proven theorems to arrive at a conclusion Slide 140 / 145 Given: 3x - 24 = 0 Prove: x = 8 Statements Reasons 1. 3x - 24 = 0 1. Given 2. 3x = 24 2. Addition 3. x = 8 3. Division Back to Instruction Pythagorean Theorem Slide 141 / 145 In a right triangle, the sum of the squares of the lengths of the legs (a and b) is equal to the square of the length of hypotenuse (c). Formula: 4 Example: 4 2 +3 2 = 5 2 16+9 = 25 25 = 25 3 5 Back to Instruction
Pythagorean Triples Slide 142 / 145 Combinations of whole numbers that work in the Pythagorean Theorem. 4 5 5 13 5 7 4 2 +3 2 = 5 2 16+9 = 25 25 = 25 3 12 5 2 +12 2 = 13 2 25+144 = 169 169 = 169 5 2 +4 2 = 7 2 25+16 = 49 41 = 49 4 Back to Instruction Right Triangle Slide 143 / 145 A triangle that has a right angle (90 ). 60º 30º 45º 45º stair case sail Back to Instruction Two-Column Proof A tool to organize your reasoning into two columns. Statements are written in the left column. Reasons are written in the right column. Slide 144 / 145 Given: 3x - 24 = 0 Prove: x = 8 Statements Reasons 1. 3x - 24 = 0 1. Given 2. 3x = 24 2. Addition 3. x = 8 3. Division Back to Instruction
Standards for Mathematical Practices Slide 145 / 145 MP1 Make sense of problems and persevere in solving them. MP2 Reason abstractly and quantitatively. MP3 Construct viable arguments and critique the reasoning of others. MP4 Model with mathematics. MP5 Use appropriate tools strategically. MP6 Attend to precision. MP7 Look for and make use of structure. MP8 Look for and express regularity in repeated reasoning. Click on each standard to bring you to an example of how to meet this standard within the unit.