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The Pythagorean Theorem c a b

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**Pythagorean Theorem Essential Questions**

How is the Pythagorean Theorem used to identify side lengths? When can the Pythagorean Theorem be used to solve real life patterns?

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**This is a right triangle:**

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**We call it a right triangle because it contains a right angle.**

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**The measure of a right angle is 90o**

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The little square in the angle tells you it is a right angle. 90o

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About 2,500 years ago, a Greek mathematician named Pythagorus discovered a special relationship between the sides of right triangles.

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**Pythagorus realized that if you have a right triangle,**

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**and you square the lengths of the two sides that make up the right angle,**

3 4 5

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and add them together, 3 4 5

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**you get the same number you would get by squaring the other side.**

3 4 5

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Is that correct? ? ?

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**It is. And it is true for any right triangle.**

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**The two sides which come together in a right angle are called**

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**The two sides which come together in a right angle are called**

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**The two sides which come together in a right angle are called**

legs.

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**The lengths of the legs are usually called a and b.**

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**The side across from the right angle is called the**

hypotenuse. a b

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**And the length of the hypotenuse is usually labeled c.**

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**The relationship Pythagorus discovered is now called The Pythagorean Theorem:**

b

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**The Pythagorean Theorem says, given the right triangle with legs a and b and hypotenuse c,**

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then c a b

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**You can use The Pythagorean Theorem to solve many kinds of problems.**

Suppose you drive directly west for 48 miles, 48

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**Then turn south and drive for 36 miles.**

48 36

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**How far are you from where you started?**

48 36 ?

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**Using The Pythagorean Theorem,**

48 482 + 362 = c2 36 c

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Why? Can you see that we have a right triangle? 48 36 c 482 362 + = c2

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**Which side is the hypotenuse?**

Which sides are the legs? 48 36 c 482 362 + = c2

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**Then all we need to do is calculate:**

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**And you end up 60 miles from where you started.**

So, since c2 is 3600, c is 60. So, since c2 is 3600, c is 48 36 60

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**Find the length of a diagonal of the rectangle:**

15" 8" ?

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**Find the length of a diagonal of the rectangle:**

15" 8" ? b = 8 c a = 15

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b = 8 a = 15 c

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**Find the length of a diagonal of the rectangle:**

15" 8" 17

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**Practice using The Pythagorean Theorem to solve these right triangles:**

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5 12 c = 13

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10 b 26

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= 24 10 b 26 (a) (c)

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Check It Out! Example 2 A rectangular field has a length of 100 yards and a width of 33 yards. About how far is it from one corner of the field to the opposite corner of the field? Round your answer to the nearest tenth.

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**Understand the Problem**

Check It Out! Example 2 Continued 1 Understand the Problem Rewrite the question as a statement. • Find the distance from one corner of the field to the opposite corner of the field. List the important information: • Drawing a segment from one corner of the field to the opposite corner of the field divides the field into two right triangles. • The segment between the two corners is the hypotenuse. • The sides of the fields are legs, and they are 33 yards long and 100 yards long.

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**Check It Out! Example 2 Continued**

Make a Plan You can use the Pythagorean Theorem to write an equation.

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**Check It Out! Example 2 Continued**

Solve 3 a2 + b2 = c2 Use the Pythagorean Theorem. = c2 Substitute for the known variables. ,000 = c2 Evaluate the powers. 11,089 = c2 Add. c Take the square roots of both sides. 105.3 c Round. The distance from one corner of the field to the opposite corner is about yards.

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**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.” a2 + b2 = c2

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Proof

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**Let’s look at it this way…**

c a c b b c2 a2 b2

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**Baseball Problem A baseball “diamond” is really a square.**

You can use the Pythagorean theorem to find distances around a baseball diamond.

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**Baseball Problem The distance between consecutive bases is 90**

feet. How far does a catcher have to throw the ball from home plate to second base?

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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? Now use: a2 + b2 = c2

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**Baseball Problem Solution**

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: x2 = = 16,200 x = ft

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Ladder Problem 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?

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**Ladder Problem Solution**

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 a2 + b2 = c2 to solve for the missing side. Distance from house: 7 meters

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**Ladder Problem Solution**

b = 24 m How did you do?

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