Slide 1 / 78 Pythagorean Theorem Distance and Midpoints Slide 2 / 78 Table of Contents Pythagorean Theorem Click on a topic to go to that section Distance Formula Midpoints Slide 3 / 78 Pythagorean Theorem Click to return to the table of contents
Slide 4 / 78 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. 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. Slide 5 / 78 Labels for a right triangle Hypotenuse click to reveal c a - Opposite the right angle click to reveal - Longest of the 3 sides b Legs click to reveal - 2 sides that form the right angle click to 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
Slide 7 / 78 Missing Leg Write Equation a 2 + b 2 = c 2 Substitute in numbers 5 2 + b 2 = 15 2 Square numbers 25 + b 2 = 225 15 ft Subtract -25 -25 Find the Square Root b 2 = 200 5 ft Label Answer Slide 8 / 78 Missing Leg a 2 + b 2 = c 2 Write Equation 18 in 9 2 + b 2 = 18 2 Substitute in numbers 9 in 81 + b 2 = 324 Square numbers Subtract -81 -81 b 2 = 243 Find the Square Root Label Answer Slide 9 / 78 Missing Hypotenuse a 2 + b 2 = c 2 Write Equation 4 2 + 7 2 = c 2 Substitute in numbers 7 in 16 + 49 = c 2 Square numbers 65 = c 2 Add 4 in Find the Square Root & Label Answer
Slide 10 / 78 How to use the formula to find missing sides. Missing Leg Missing Hypotenuse Write Equation Write Equation Substitute in numbers Substitute in numbers Square numbers Square numbers Subtract Add Find the Square Root Find the Square Root Label Answer Label Answer Slide 11 / 78 1 What is the length of the third side? Answer x 7 4 Slide 12 / 78 2 What is the length of the third side? Answer x 41 15
Slide 13 / 78 3 What is the length of the third side? Answer 7 4 z Slide 14 / 78 4 What is the length of the third side? Answer x 3 4 Slide 15 / 78 Pythagorean Triplets There are combinations of whole numbers that work in the Pythagorean Theorem. 5 3 These sets of numbers are known as Pythagorean Triplets. 3-4-5 is the most famous of the triplets. If you recognize 4 the sides of the triangle as being a triplet (or multiple of one), you won't need a calculator!
Slide 16 / 78 Can you find any other Pythagorean Triplets? Use the list of squares to see if any other triplets work. 1 2 = 1 11 2 = 121 21 2 = 441 Triples 2 2 = 2 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 = 400 10 2 = 100 20 2 = 400 30 2 = 900 Slide 17 / 78 5 What is the length of the third side? Answer 6 8 Slide 18 / 78 6 What is the length of the third side? Answer 5 13
Slide 19 / 78 7 What is the length of the third side? Answer 48 50 Slide 20 / 78 8 The legs of a right triangle are 7.0 and Answer 3.0, what is the length of the hypotenuse? Slide 21 / 78 9 The legs of a right triangle are 2.0 and 12, what is the length of the Answer hypotenuse?
Slide 22 / 78 10 The hypotenuse of a right triangle has a length of 4.0 and one of its legs has a Answer length of 2.5. What is the length of the other leg? Slide 23 / 78 11 The hypotenuse of a right triangle has a Answer length of 9.0 and one of its legs has a length of 4.5. What is the length of the other leg? Slide 24 / 78 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. c = 5 ft a = 3 ft b = 4 ft
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. Slide 26 / 78 8 in, 17 in, 15 in Is it a Right Triangle? a 2 + b 2 = c 2 Write Equation 8 2 + 15 2 = 17 2 Plug in numbers 64 + 225 = 289 Square numbers 289 = 289 Simplify both sides Yes! Are they equal? Slide 27 / 78 12 Is the triangle a right triangle? Answer Yes No 10 ft 6 ft 8 ft
Slide 28 / 78 13 Is the triangle a right triangle? Answer Yes No 36 ft 24 ft 30 ft Slide 29 / 78 14 Is the triangle a right triangle? Yes 10 in. Answer No 8 in. 12 in. Slide 30 / 78 15 Is the triangle a right triangle? Answer Yes No 13 ft 5 ft 12 ft
Slide 31 / 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 Answer No Slide 32 / 78 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 17 The sizes of television and computer monitors are Answer 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 34 / 78 18 A tree was hit by lightning during a storm. The part of Answer 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 19 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 Answer diamond is a square) 2nd 90 ft. 90 ft. 1st 3rd 90 ft. 90 ft. home Slide 36 / 78 20 You're locked out of your house and the only open window is on the second floor, 25 feet above ground. Answer 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 37 / 78 Slide 38 / 78 Distance Formula Click to return to the table of contents Slide 39 / 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. The distance between these two points is 4. The top point is 4 above the lower point.
Slide 40 / 78 22 What is the distance between these two points? Pull Pull Slide 41 / 78 23 What is the distance between these two points? Slide 42 / 78 24 What is the distance between these two points?
Slide 43 / 78 Most sets of points do not lie in a vertical or horizontal line. For example: 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. Slide 44 / 78 Draw the right triangle around these two points. Then use the Pythagorean theorem to find the distance in red. c 2 = a 2 + b 2 c 2 = 3 2 + 4 2 c c 2 = 9 + 16 b c 2 = 25 c = 5 a The distance between the two points (2,2) and (5,6) is 5 units. Slide 45 / 78 Example: c 2 = a 2 + b 2 c 2 = 3 2 + 6 2 c 2 = 9 + 36 c 2 = 45 The distance between the two points (-3,8) and (-9,5) is approximately 6.7 units.
Slide 46 / 78 Try This: c 2 = a 2 + b 2 c 2 = 9 2 + 12 2 c 2 = 81 + 144 c 2 = 225 c = 15 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 c 2 = a 2 + b 2 Create a right triangle around the two points. Label the points as shown. d 2 = (x 2 - x 1 ) 2 + (y 2 - y 1 ) 2 Then substitute into the Pythagorean Formula. d = (x 2 - x 1 ) 2 + (y 2 - y 1 ) 2 This is the distance formula now substitute in (x 2 , y 2 ) values. length = d d = (5 - 2) 2 + (6 - 2) 2 y 2 - y 1 (x 1 , y 1 ) d = (3) 2 + (4) 2 length = x 2 - x 1 d = 9 + 16 d = 25 d = 5
Slide 49 / 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 50 / 78 Slide 51 / 78
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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? C (9,4) D (3,3) B (8,0) A (0,-1) Slide 57 / 78
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