Pythagorean Theorem

Foldable 1.Cut out the rectangle. Fold down the blank to rectangle, and fold up the blank bottom rectangle. Cut only the lines separating the blank pieces. 2.Glue the center piece into the notebook.

Foldable Continued… 1.Label the flaps as seen in the picture to the left. The first tabs are “words.” The middle tabs are “formula.” The last tab is “in action.”

Right Triangle Hypotenuse Legs Pythagorean Theorem Radical/Radicand Square Root VOCABULARY

RIGHT TRIANGLE Longest side is the hypotenuse, side c (opposite the 90 o angle) The other two sides are the legs, sides a and b Pythagoras developed a formula for finding the length of the sides of any right triangle

RADICAL/RADICAND

Definition: A number the produces a specified quantity when multiplied by itself. SQUARE ROOT

THE PYTHAGOREAN THEOREM “For any right triangle, the sum of the areas of the two small squares is equal to the area of the larger.” a 2 + b 2 = c 2 a 2 + b 2 = c 2 In different words… a 2 + b 2 = c 2 If the angle opposite the hypotenuse is a right angle, then a 2 + b 2 = c 2

In Words: C 2 – B 2 = A 2 C 2 – A 2 = B 2 If a triangle is a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse of the triangle. FOLDABLE EXAMPLE – PYTHAGOREAN THEOREM FORMULA: A 2 + B 2 = C 2

FOLDABLE EXAMPLE: IN ACTION COPY THE TRIANGLE AND MEASURES ON YOUR “IN ACTION” TAB. WE WILL WORK TOGETHER.

YOUR TURN! 

Normal: If x, then y. Converse: If y, then x. Example: If it is raining, then the grass is wet. Converse: If the grass is wet, then it is raining. CONVERSE?

a 2 + b 2 = c 2 “If a triangle has sides of lengths a, b, and c; and a 2 + b 2 = c 2 then the triangle is a right triangle with hypotenuse of length c.” CONVERSE OF THE PYTHAGOREAN THEOREM You can determine if a triangle is a right triangle if a 2 + b 2 = c 2 the 3 side lengths fit into a 2 + b 2 = c 2 COPY THE STATEMENT IN THE BOX BELOW ON YOUR “WORDS” TAB FOR THE CONVERSE OF THE PYTHAGOREAN THEOREM.

FOLDABLE EXAMPLE: FORMULA COPY THE FOLLOWING INTO YOUR “FORMULA” BOX. Is a triangle with side lengths 11, 6, 5 a right triangle? A 2 + B 2 = C = = ≠ 121 No, it is not a right triangle!!!

FOLDABLE EXAMPLE: IN ACTION COPY THE FOLLOWING INTO YOUR “IN ACTION” BOX. Is a triangle with side lengths 12, 20, 16 a right triangle? A 2 + B 2 = C = = = 400 YES, it is a right triangle!!!

YOUR TURN! 

COMMON PYTHAGOREAN TRIPLES

Find the length of the hypotenuse if 1. a = 12 and b = = c = c = c 2 Take the square root of both sides. 20 = c

Find the length of the hypotenuse if 2. a = 5 and b = 7.

Find the length of the hypotenuse given a = 6 and b =

Find the length of the leg, to the nearest hundredth, if 3. a = 4 and c = 10.

Find the length of the leg, to the nearest hundredth, if 4. c = 10 and b = 7.

Find the length of the missing side given a = 4 and c =

5. The measures of three sides of a triangle are given below. Determine whether each triangle is a right triangle. 5, 3, and 8

The rectangular bottom of a box has an interior length of 19 inches and an interior width of 14 inches. A stick is placed in the box along the diagonal of the bottom of the box. Which measurement is closest to the longest possible length for the stick? a 2 + b 2 = c 2 F 12.8 in. G 16.3 in. H 23.6 in. J 33.0 in.

The Pythagorean theorem has far-reaching ramifications in other fields (such as the arts), as well as practical applications. The theorem is invaluable when computing distances between two points, such as in navigation and land surveying. Another important application is in the design of ramps. Ramp designs for handicap-accessible sites and for skateboard parks are very much in demand. Can you think of other ways the Pythagorean Theorem can be extremely valuable? REAL-LIFE APPLICATIONS

Baseball Problem A baseball “diamond” is really a square. You can use the Pythagorean theorem to find distances around a baseball diamond.

The distance between consecutive bases is 90 feet. How far does a catcher have to throw the ball from home plate to second base? 28 BASEBALL PROBLEM

To use the Pythagorean theorem to solve for x, find the right angle. Which side is the hypotenuse? Which sides are the legs? a 2 + b 2 = c 2 Now use: a 2 + b 2 = c 2 29 BASEBALL PROBLEM

The hypotenuse is the distance from home to second, or side x in the picture. The legs are from home to first and from first to second. Solution:  x 2 = = 16,200  x = ft 30 BASEBALL PROBLEM SOLUTION

 A ladder leans against a second-story window of a house. If the ladder is 25 meters long, and the base of the ladder is 7 meters from the house, how high is the window? 31 LADDER PROBLEM

First draw a diagram that shows the sides of the right triangle. Label the sides:  Ladder is 25 m  Distance from house is 7 m Use a 2 + b 2 = c 2 to solve for the missing side. 32 LADDER PROBLEM SOLUTION Distance from house: 7 meters

 b 2 = 25 2  49 + b 2 = 625  b 2 = 576  b = 24 m  How did you do? 33 LADDER PROBLEM SOLUTION B

EXAMPLE 1

EXAMPLE 2

EXAMPLE 3 Laramie leaves the trailhead and hikes 5 miles east and 9 miles north to a waterfall. What is the closest distance from the trailhead to the waterfall? Draw a picture to help solve.

EXAMPLE 4 Hailey uses a straight piece of wood that is 8 feet long to prop up an old fence. If the fence is 6 feet tall, how far from the fence is the bottom of the piece of wood? Draw a picture to help solve.

EXAMPLE 5 A right triangle in the coordinate plane has vertices at (0, 6), (8, 0), and (0, 0). What is the length of the hypotenuse?

EXAMPLE 6 To the nearest tenth, what is the distance from R to T?

EXAMPLE 7

EXAMPLE 8 The triangle on the grid represents a section of Miesha’s backyard. What is the length of the hypotenuse?

42 DO NOT DO THIS!