Pythagorean Theorem Distance and Midpoints

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Slide 1 / 78 Pythagorean Theorem Distance and Midpoints Slide 2 / 78 Table of Contents Pythagorean Theorem Distance Formula Midpoints Click on a topic to go to that section Slide 3 / 78 Slide 4 / 78

1 Slide 1 / 78 Pythagorean Theorem Distance and Midpoints Slide 2 / 78 Table of Contents Pythagorean Theorem Distance Formula Midpoints Click on a topic to go to that section Slide 3 / 78 Slide 4 / 78 Pythagorean Theorem Pythagorean Theorem This is a theorem that 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. Click to return to the table of contents 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 Slide 5 / 78 Labels for a right triangle b c Hypotenuse click to reveal - Opposite the right angle - Longest of click the to 3 sides reveal Slide 6 / 78 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 Link to animation of proof Legs click to reveal - 2 sides that click form to the reveal right angle

Slide 7 / 78 Slide 8 / 78 Missing Leg Missing Leg a 2 + b 2 = c 2 Write Equation a 2 + b 2 = c 2 Write Equation 15 ft 5 2 + b 2 = 15 2 25 + b 2 = 225-25 -25 Substitute in numbers Square numbers

2 Slide 7 / 78 Slide 8 / 78 Missing Leg Missing Leg a 2 + b 2 = c 2 Write Equation a 2 + b 2 = c 2 Write Equation 15 ft b 2 = b 2 = Substitute in numbers Square numbers Subtract 9 in 18 in b 2 = b 2 = 324 Substitute in numbers Square numbers 5 ft b 2 = 200 Find the Square Root Label b 2 = 243 Subtract Find the Square Root Label Slide 9 / 78 Missing Hypotenuse Slide 10 / 78 How to use the formula to find missing sides. a 2 + b 2 = c 2 Write Equation Missing Leg Missing Hypotenuse 7 in = c = c 2 Substitute in numbers Square numbers Write Equation Substitute in numbers Square numbers Write Equation Substitute in numbers Square numbers 4 in 65 = c 2 Add Find the Square Root & Label Subtract Find the Square Root Label Add Find the Square Root Label Slide 11 / 78 1 What is the length of the third side? Slide 12 / 78 2 What is the length of the third side? 7 x x

3 Slide 13 / 78 3 What is the length of the third side? Slide 14 / 78 4 What is the length of the third side? z x Slide 15 / 78 Pythagorean Triplets Slide 16 / 78 Can you find any other Pythagorean Triplets? There are combinations of whole numbers that work in the Pythagorean Theorem. These sets of numbers are known as Pythagorean Triplets is the most famous of the triplets. If you recognize the sides of the triangle as being a triplet (or multiple of one), you won't need a calculator! Triples Use the list of squares to see if any other triplets work. 1 2 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 900 Slide 17 / 78 Slide 18 / 78 5 What is the length of the third side? 6 What is the length of the third side?

4 Slide 19 / 78 Slide 20 / 78 7 What is the length of the third side? 48 8 The legs of a right triangle are 7.0 and 3.0, what is the length of the hypotenuse? 50 Slide 21 / 78 Slide 22 / 78 9 The legs of a right triangle are 2.0 and 12, what is the length of the hypotenuse? 10 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 23 / 78 Slide 24 / 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? Corollary to the 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 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. a = 3 ft c = 5 ft b = 4 ft

5 Slide 25 / 78 Corollary to the Pythagorean Theorem 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. 8 in, 17 in, 15 in a 2 + b 2 = c = = = 289 Yes! Slide 26 / 78 Is it a Right Triangle? Write Equation Plug in numbers Square numbers Simplify both sides Are they equal? Slide 27 / 78 Slide 28 / Is the triangle a right triangle? Yes No 6 ft 10 ft 13 Is the triangle a right triangle? Yes No 36 ft 24 ft 30 ft 8 ft Slide 29 / Is the triangle a right triangle? Slide 30 / Is the triangle a right triangle? Yes No 8 in. 10 in. 12 in. Yes No 5 ft 12 ft 13 ft

Slide 31 / 78 Slide 32 / 78 16 Can you construct a right triangle with three lengths of wood that measure 7.5 in, 18 in and 19.5 in? Yes No Steps to Pythagorean Theorem Application Problems. 1. Draw a right triangle to represent the situation.

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?

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?

6 Slide 31 / 78 Slide 32 / Can you construct a right triangle with three lengths of wood that measure 7.5 in, 18 in and 19.5 in? Yes No 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. Slide 33 / 78 Slide 34 / 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? 18 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 35 / 78 Slide 36 / 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) 2nd 20 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? 90 ft. 90 ft. 3rd 1st 90 ft. 90 ft. home

7 Slide 37 / 78 Slide 38 / 78 Distance Formula Click to return to the table of contents Slide 39 / 78 Slide 40 / 78 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. 22 What is the distance between these two points? Pull The distance between these two points is 4. The top point is 4 above the lower point. Slide 41 / What is the distance between these two points? Slide 42 / What is the distance between these two points?

8 Slide 43 / 78 Most sets of points do not lie in a vertical or horizontal line. For example: Slide 44 / 78 Draw the right triangle around these two points. Then use the Pythagorean theorem to find the distance in red. 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. c a b c 2 = a 2 + b 2 c 2 = c 2 = c 2 = 25 c = 5 The distance between the two points (2,2) and (5,6) is 5 units. Slide 45 / 78 Slide 46 / 78 Example: Try This: c 2 = a 2 + b 2 c 2 = c 2 = c 2 = 45 c 2 = a 2 + b 2 c 2 = c 2 = c 2 = 225 c = 15 The distance between the two points (-3,8) and (-9,5) is approximately 6.7 units. The distance between the two points (-5, 5) and (7, -4) is 15 units. Slide 47 / 78 Deriving a formula for calculating distance... Slide 48 / 78 Create a right triangle around the two points. Label the points as shown. Then substitute into the Pythagorean Formula. d (x 1, y 1) (x 2, y 2) length = y 2 - y 1 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. d = (5-2) 2 + (6-2) 2 d = (3) 2 + (4) 2 length = x 2 - x 1 d = d = 25 d = 5

9 Slide 49 / 78 Slide 50 / 78 Distance Formula 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 Slide 51 / 78 Slide 52 / 78 Slide 53 / 78 Slide 54 / 78

Slide 55 / 78 How would you find the perimeter of this rectangle? Either just count the units or find the distance between the points from the ordered pairs.

10 Slide 55 / 78 How would you find the perimeter of this rectangle? Either just count the units or find the distance between the points from the ordered pairs. Slide 56 / 78 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) Slide 57 / 78 Slide 58 / 78 Slide 59 / 78 Slide 60 / 78

11 Slide 61 / 78 Slide 62 / 78 Click to return to the table of contents Find the midpoint of the line segment. What is a midpoint? How did you find the midpoint? What are the coordinates of the midpoint? Midpoints (2, 10) (2, 2) Slide 63 / 78 Find the midpoint of the line segment. What are the coordinates of the midpoint? How is it related to the coordinates of the endpoints? Slide 64 / 78 Find the midpoint of the line segment. What are the coordinates of the midpoint? How is it related to the coordinates of the endpoints? (3, 4) (9, 4) (3, 4) (9, 4) Midpoint = (6, 4) It is in the middle of the segment. Average of x-coordinates. Average of y-coordinates. Slide 65 / 78 Slide 66 / 78 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: The midpoint of a segment AB is the point M on AB halfway between the endpoints A and B. A (2,5) ( x1 + x 2 y 1 + y 2, 2 2 ) B (8,1) The x and y coordinates of the midpoint are the averages of the x and y coordinates of the endpoints, respectively. See next page for answer

$Write fractions in simplest form: (5,3) is the midpoint of AB Find the midpoint of (1,0) and (-5,3) Use the midpoint formula: + x 2 y 1 + y 2, 2 2 ) ( x1$

12 Slide 67 / 78 Slide 68 / 78 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 Substitute in values: 2 + 8, ( 2 2 ) Simplify the numerators: 10, 6 ( 2 2 ) Write fractions in simplest form: (5,3) is the midpoint of AB Find the midpoint of (1,0) and (-5,3) Use the midpoint formula: + x 2 y 1 + y 2, 2 2 ) ( x1 Substitute in values: , ( 2 2 ) Simplify the numerators: -4, 3 ( 2 2 ) Write fractions in simplest form: (-2,1.5) is the midpoint Slide 69 / 78 Slide 70 / 78 Slide 71 / 78 Slide 72 / 78

13 Slide 73 / 78 Slide 74 / Find the center of the circle with a diameter having endpoints at (-4,3) and (0,2). Which formula should be used to solve this problem? A B C D Pythagorean Formula Distance Formula Midpoint Formula Formula for Area of a Circle Slide 75 / 78 Slide 76 / 78 Slide 77 / 78 Slide 78 / 78

2.10 Theorem of Pythagoras

2.10 Theorem of Pythagoras Dr. Robert J. Rapalje, Retired Central Florida, USA Before introducing the Theorem of Pythagoras, we begin with some perfect square equations. Perfect square equations (see the