Pythagorean Theorem Distance Formula Distance and Midpoints - - PDF document

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Pythagorean Theorem Distance and Midpoints

Slide 2 / 78 Distance Formula Midpoints Table of Contents

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Pythagorean Theorem Slide 3 / 78 Pythagorean Theorem

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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 Legs

• Opposite the right angle
• Longest of the 3 sides
• 2 sides that form the right angle

click to reveal click to reveal

click to reveal

Labels for a right triangle c

a b

Hypotenuse

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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).

a2 + b2 = c2

Link to animation of proof

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a2 + b2 = c2 52 + b2 = 152 25 + b2 = 225

• 25 -25

b2 = 200 Missing Leg Write Equation Substitute in numbers Square numbers Subtract Find the Square Root Label Answer

5 ft 15 ft

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9 in 18 in

a2 + b2 = c2 92 + b2 = 182 81 + b2 = 324

• 81 -81

b2 = 243 Missing Leg Write Equation Substitute in numbers Square numbers Subtract Find the Square Root Label Answer

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4 in 7 in

a2 + b2 = c2 42 + 72 = c2 16 + 49 = c2 65 = c2 Missing Hypotenuse Write Equation Substitute in numbers Square numbers Add Find the Square Root & Label Answer

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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 How to use the formula to find missing sides.

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1 What is the length of the third side?

4 7 x

Answer

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2 What is the length of the third side?

Answer

41 x 15

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3 What is the length of the third side?

Answer

7 z 4

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4 What is the length of the third side?

3 4 x

Answer

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3 4

There are combinations of whole numbers that work in the Pythagorean Theorem. These sets of numbers are known as Pythagorean Triplets. 3-4-5 is the most famous of the triplets. If you recognize the sides of the triangle as being a triplet (or multiple of

• ne), you won't need a

calculator!

5

Pythagorean Triplets Slide 16 / 78

Triples

12 = 1 112 = 121 212 = 441 22 = 2 122 = 144 222 = 484 32 = 9 132 = 169 232 = 529 42 = 16 142 = 196 242 = 576 52 = 25 152 = 225 252 = 625 62 = 36 162 = 256 262 = 676 72 = 49 172 = 289 272 = 729 82 = 64 182 = 324 282 = 784 92 = 81 192 = 361 292 = 400 102 = 100 202 = 400 302 = 900 Can you find any other Pythagorean Triplets? Use the list of squares to see if any other triplets work.

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5 What is the length of the third side?

6 8

Answer

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6 What is the length of the third side?

5 13

Answer

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7 What is the length of the third side?

48 50

Answer

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8 The legs of a right triangle are 7.0 and 3.0, what is the length of the hypotenuse?

Answer

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9 The legs of a right triangle are 2.0 and 12, what is the length of the hypotenuse?

Answer

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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

• ther leg?

Answer

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11 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

• ther leg?

Answer

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 c2 = a2 + b2, then the triangle is a right triangle. If c2 ≠ a2 + b2, then the triangle is not a right triangle.

b = 4 ft c = 5 ft a = 3 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.

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Is it a Right Triangle? Write Equation Plug in numbers Square numbers Simplify both sides Are they equal?

8 in, 17 in, 15 in a2 + b2 = c2 82 + 152 = 172 64 + 225 = 289 289 = 289 Yes!

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12 Is the triangle a right triangle? Yes No

8 ft 10 ft 6 ft

Answer

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13 Is the triangle a right triangle? Yes No

30 ft 24 ft 36 ft

Answer

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14 Is the triangle a right triangle? Yes No

10 in. 8 in. 12 in.

Answer

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15 Is the triangle a right triangle? Yes No

Answer

5 ft 13 ft 12 ft

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Answer

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

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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.

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17 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

• f the screen?

Answer

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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?

Answer

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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 diamond is a square) home 1st 2nd 3rd 90 ft. 90 ft. 90 ft. 90 ft.

Answer

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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?

Answer

Slide 37 / 78 Slide 38 / 78 Distance Formula

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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.

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22 What is the distance between these two points?

Pull Pull

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23 What is the distance between these two points?

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24 What is the distance between these two points?

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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.

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Draw the right triangle around these two points. Then use the Pythagorean theorem to find the distance in red. c2 = a2 + b2 c2 = 32 + 42 c2 = 9 + 16 c2 = 25 c = 5 a b c The distance between the two points (2,2) and (5,6) is 5 units.

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Example: c2 = a2 + b2 c2 = 32 + 62 c2 = 9 + 36 c2 = 45 The distance between the two points (-3,8) and (-9,5) is approximately 6.7 units.

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Try This: c2 = a2 + b2 c2 = 92 + 122 c2 = 81 + 144 c2 = 225 c = 15 The distance between the two points (-5, 5) and (7, -4) is 15 units.

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Deriving a formula for calculating distance...

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Create a right triangle around the two

• points. Label the points as shown.

Then substitute into the Pythagorean Formula. (x1, y1) length = x2 - x1 length = y2 - y1 d c2 = a2 + b2 d2 = (x2 - x1)2 + (y2 - y1)2 d = (x2 - x1)2 + (y2 - y1)2 This is the distance formula now substitute in values. d = (5 - 2)2 + (6 - 2)2 d = (3)2 + (4)2 d = 9 + 16 d = 25 d = 5 (x2, y2)

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Distance Formula

d = (x2 - x1)2 + (y2 - y1)2

You can find the distance d between any two points (x1, y1) and (x2, y2) using the formula below.

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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.

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A (0,-1) B (8,0) C (9,4) D (3,3) Can we just count how many units long each line segment is in this quadrilateral to find the perimeter?

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Slide 61 / 78 Midpoints

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(2, 2) (2, 10) Find the midpoint of the line segment. What is a midpoint? How did you find the midpoint? What are the coordinates of the midpoint?

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(3, 4) (9, 4) Find the midpoint of the line segment. What are the coordinates of the midpoint? How is it related to the coordinates of the endpoints?

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(3, 4) (9, 4) Find the midpoint of the line segment. What are the coordinates of the midpoint? How is it related to the coordinates of the endpoints? Midpoint = (6, 4) It is in the middle of the segment. Average of x-coordinates. Average of y-coordinates.

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The Midpoint Formula To calculate the midpoint of a line segment with endpoints (x1,y1) and (x2,y2) use the formula:

(

x1 + x2 y1 + y2 2 2 ,

)

The x and y coordinates of the midpoint are the averages

• f the x and y coordinates of the endpoints, respectively.

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The midpoint of a segment AB is the point M on AB halfway between the endpoints A and B. B (8,1) A (2,5) See next page for answer

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The midpoint of a segment AB is the point M on AB halfway between the endpoints A and B. B (8,1) A (2,5) Use the midpoint formula: Substitute in values:

(

x1 + x2 y1 + y2 2 2 ,

)

2 + 8 , 5 + 1 2 2

( )

Simplify the numerators: 10 6 2 2 , Write fractions in simplest form:

( )

(5,3) is the midpoint of AB M

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Find the midpoint of (1,0) and (-5,3) Use the midpoint formula: Substitute in values:

(

x1 + x2 y1 + y2 2 2 ,

)

1 + -5 , 0 + 3 2 2

( )

Simplify the numerators:

• 4

3 2 2 , Write fractions in simplest form:

( )

(-2,1.5) is the midpoint

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36 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 Pythagorean Formula B Distance Formula C Midpoint Formula D Formula for Area of a Circle

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