Pythagorean Theorem Distance Formula Distance and Midpoints - PDF document

Slide 1 / 78 Slide 2 / 78 Table of Contents Pythagorean Theorem Click on a topic to go to that section Pythagorean Theorem Distance Formula Distance and Midpoints Midpoints 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. Pythagoras lived during the 6th century B.C. on the island of Samos in the Aegean Sea. He also lived in Egypt, Babylon, Click to return to the table of contents and southern Italy. He was a philosopher and a teacher. Slide 5 / 78 Slide 6 / 78 Labels for a right triangle 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 ). Hypotenuse click to reveal c a - Opposite the right angle click to reveal a 2 + b 2 = c 2 - Longest of the 3 sides Link to animation of proof b Legs click to reveal - 2 sides that form the right angle click to reveal

Slide 7 / 78 Slide 8 / 78 Missing Leg Missing Leg Write Equation a 2 + b 2 = c 2 a 2 + b 2 = c 2 Write Equation Substitute in numbers 5 2 + b 2 = 15 2 18 in 9 2 + b 2 = 18 2 Substitute in numbers 9 in Square numbers 25 + b 2 = 225 15 ft 81 + b 2 = 324 Square numbers Subtract -25 -25 Subtract -81 -81 b 2 = 200 Find the Square Root 5 ft Label Answer b 2 = 243 Find the Square Root Label Answer Slide 9 / 78 Slide 10 / 78 Missing Hypotenuse How to use the formula to find missing sides. Write Equation a 2 + b 2 = c 2 Missing Leg Missing Hypotenuse 4 2 + 7 2 = c 2 Write Equation Write Equation Substitute in numbers 7 in Substitute in numbers Substitute in numbers 16 + 49 = c 2 Square numbers Square numbers Square numbers 65 = c 2 Subtract Add Add 4 in Find the Square Root Find the Square Root Find the Square Root & Label Answer Label Answer Label Answer Slide 11 / 78 Slide 12 / 78 1 What is the length of the third side? 2 What is the length of the third side? Answer Answer x x 7 41 4 15

Slide 13 / 78 Slide 14 / 78 3 What is the length of the third side? 4 What is the length of the third side? Answer Answer 7 x 3 4 z 4 Slide 15 / 78 Slide 16 / 78 Pythagorean Triplets Can you find any other Pythagorean Triplets? Use the list of squares to see if any other There are combinations of triplets work. whole numbers that work in the Pythagorean Theorem. 5 3 These sets of numbers are 1 2 = 1 11 2 = 121 21 2 = 441 Triples known as Pythagorean 2 2 = 2 12 2 = 144 22 2 = 484 Triplets. 3 2 = 9 13 2 = 169 23 2 = 529 4 2 = 16 14 2 = 196 24 2 = 576 3-4-5 is the most famous of 5 2 = 25 15 2 = 225 25 2 = 625 the triplets. If you recognize 6 2 = 36 16 2 = 256 26 2 = 676 4 the sides of the triangle as 7 2 = 49 17 2 = 289 27 2 = 729 being a triplet (or multiple of 8 2 = 64 18 2 = 324 28 2 = 784 one), you won't need a 9 2 = 81 19 2 = 361 29 2 = 400 calculator! 10 2 = 100 20 2 = 400 30 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? Answer Answer 5 6 13 8

Slide 19 / 78 Slide 20 / 78 7 What is the length of the third side? 8 The legs of a right triangle are 7.0 and Answer 3.0, what is the length of the Answer hypotenuse? 48 50 Slide 21 / 78 Slide 22 / 78 10 The hypotenuse of a right triangle has a 9 The legs of a right triangle are 2.0 and length of 4.0 and one of its legs has a Answer 12, what is the length of the Answer length of 2.5. What is the length of the hypotenuse? other leg? Slide 23 / 78 Slide 24 / 78 11 The hypotenuse of a right triangle has a Corollary to the Pythagorean Theorem Answer length of 9.0 and one of its legs has a length of 4.5. What is the length of the If a and b are measures of the shorter sides of a triangle, other leg? 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 Slide 26 / 78 Corollary to the Pythagorean Theorem 8 in, 17 in, 15 in Is it a Right Triangle? a 2 + b 2 = c 2 Write Equation In other words, you can check to see if a triangle is a right triangle by seeing if the Pythagorean Theorem is true. 8 2 + 15 2 = 17 2 Plug in numbers Test the Pythagorean Theorem. If the final equation is true, 64 + 225 = 289 then the triangle is right. If the final equation is false, then the Square numbers triangle is not right. 289 = 289 Simplify both sides Yes! Are they equal? Slide 27 / 78 Slide 28 / 78 12 Is the triangle a right triangle? 13 Is the triangle a right triangle? Answer Answer Yes Yes No No 36 ft 10 ft 24 ft 6 ft 30 ft 8 ft Slide 29 / 78 Slide 30 / 78 14 Is the triangle a right triangle? 15 Is the triangle a right triangle? Answer Yes Yes 10 in. Answer No No 13 ft 5 ft 8 in. 12 in. 12 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 Answer Steps to Pythagorean Theorem Application Problems. No 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 / 78 18 A tree was hit by lightning during a storm. The part of 17 The sizes of television and computer monitors are Answer Answer the tree still standing is 3 meters tall. The top of the given in inches. However, these dimensions are tree is now resting 8 meters from the base of the tree, actually the diagonal measure of the rectangular and is still partially attached to its trunk. Assume the screens. Suppose a 14-inch computer monitor has an ground is level. How tall was the tree originally? actual screen length of 11-inches. What is the height of the screen? Slide 35 / 78 Slide 36 / 78 19 You've just picked up a ground ball at 3rd base, and you 20 You're locked out of your house and the only open see the other team's player running towards 1st base. window is on the second floor, 25 feet above ground. Answer How far do you have to throw the ball to get it from third There are bushes along the edge of your house, so base to first base, and throw the runner out? (A baseball Answer you'll have to place a ladder 10 feet from the house. diamond is a square) 2nd What length of ladder do you need to reach the window? 90 ft. 90 ft. 1st 3rd 90 ft. 90 ft. home

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 22 What is the distance between these two points? find the distance between them by simply counting units on the graph, since they lie in a vertical line. Pull Pull The distance between these two points is 4. The top point is 4 above the lower point. Slide 41 / 78 Slide 42 / 78 23 What is the distance between these two points? 24 What is the distance between these two points?

Slide 43 / 78 Slide 44 / 78 Most sets of points do not lie in a vertical or horizontal line. Draw the right triangle around these two points. Then use the For example: Pythagorean theorem to find the distance in red. c 2 = a 2 + b 2 Counting the units c 2 = 3 2 + 4 2 between these two c c 2 = 9 + 16 b points is impossible. c 2 = 25 So mathematicians c = 5 a have developed a formula using the Pythagorean theorem to find the distance The distance between two points. 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 = a 2 + b 2 c 2 = 3 2 + 6 2 c 2 = 9 2 + 12 2 c 2 = 9 + 36 c 2 = 81 + 144 c 2 = 45 c 2 = 225 c = 15 The distance between The distance between the two points (-5, 5) the two points (-3,8) and (-9,5) is and (7, -4) is 15 units. approximately 6.7 units. Slide 47 / 78 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 Deriving a formula for calculating 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 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 Slide 56 / 78 How would you find the perimeter of this rectangle? Can we just count how many units long each line segment is in this quadrilateral to find the perimeter? Either just count the units or find the distance between the points from the ordered C (9,4) D (3,3) pairs. B (8,0) A (0,-1) Slide 57 / 78 Slide 58 / 78 Slide 59 / 78 Slide 60 / 78