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Hi, my name is Carl the Calculator and I’ll be your partner for the day. Throughout this presentation, there will be buttons at the bottom of each slide to help you navigate. Let’s begin by clicking the NEXT button. NextBack 1

The Pythagorean Theorem: Who was Pythagoras? Pythagoras was born on a small island, Samos, just off the east coast of Greece. He lived from B.C. He traveled extensively in his youth and many historians attribute his mathematical ability to the knowledge he gained in his travels. 2 3 NextBack

“In order to understand how Pythagoras discovered his famous formula, we first need to review how to find the areas of triangles and squares.” Back Review of Areas Skip Review 1

Area of Squares “To find the area of a square, you need to multiply the side of the square by itself. So the area of the square given below is a x a or a 2.” a a a a Back Go to Quiz Skip Review Area = a x a = a 2 1

Quiz: Area of Squares What is the area of the square given below (click on the correct letter): a)2020 b)3030 c)2525 d) Back Skip Review

Quiz: Area of Squares “Yah! You got it right! Now let’s see how you do with triangles.” Back Skip Review Go to Triangles 1

Quiz: Area of Squares “Ah shucks, that’s not the right answer. The area of the square is 5 x 5 = 25. You can go back and review squares again or go on the triangles.” Back Skip Review Go to Triangles Back to Squares 1

Area of Triangles “To find the area of a triangle, you multiply the base by the height and divide by 2. So the area of the triangle given below is (a x b)/2.” Area = (a x b)/2 Back Go to Quiz Skip Review 1

Quiz: Area of Triangles What is the area of the triangle given below? a)2020 b)99 c)1010 d)1515 Back Skip Review 5 4

Quiz: Area of Triangles “Oh Yeah! That’s right! Looks like you’ve mastered areas. Now let’s move on to the Pythagorean Theorem.” Back Pythagorean Theorem 1

Quiz: Area of Triangles “Oh, so sorry! That’s not the right answer. The area of the triangle is (4 x 5)/2 = 10. You can go back and review triangles again, or go on to the Pythagorean Theorem!” Back Pythagorean Theorem Back to Triangles 1

What is the Pythagorean Theorem? “The Pythagorean theorem is a formula that describes the relationship between the sides in a right triangle. A right triangle is a triangle with a 90⁰ angle, like the one below.” NextBack 1

What is this relationship? “The Pythagorean Theorem states that the square of the hypotenuse is equal to the sum of the squares of the sides. The hypotenuse is the side opposite the right angle. So, in the triangle below, c 2 = a 2 + b 2.” NextBack Right Angle Hypotenuse 1

How do we know this is true? Look at the figure below: Notice that there are two different squares: the outside square has sides of length (a + b) and the inside square has sides of length c. 4 NextBack

Also notice that there are 4 right triangles with sides of a and b and with a hypotenuse of c. And that the area of the outside square is (a+b) 2 which is a 2 + 2ab + b 2. NextBack 4

The area of the outside square can also be thought of as the area of the inside square plus the area of the 4 triangles. So this can be written as: c 2 + 4(ab)/2 = c 2 + 2ab NextBack 4

“Now we have two expressions for the same area, so we can set them equal to each other: a 2 + 2ab + b 2 = c 2 + 2ab. Canceling 2ab from both sides reveals a 2 + b 2 = c 2, which is the Pythagorean Theorem.” Back Restart Proof Go to Application 1

“Now let’s reflect on what we’ve learned: With what kind of triangles can we apply the Pythagorean Theorem? With what kind of triangles can we apply the Pythagorean Theorem? Could we use it with this triangle? How do you know which side of the triangle is the hypotenuse? How do you know which side of the triangle is the hypotenuse? If you are unsure of how to answer one of these questions, click on the question to find the answer. Otherwise, move on to the evaluation.” Back Go to Evaluation 1

With what kind of triangles can we apply the Pythagorean Theorem? “You can only use the Pythagorean Theorem with right triangles. A right triangle is a triangle with a right angle (an angle which measures 90 0.” Right Triangle Back Go to Evaluation 1

Could we use it with this triangle? “So, we know that we can only apply the Pythagorean Theorem to right triangles. So the question is whether or not this is a right triangle. And, if we look closely, this triangle does not have a right angle so it is not a right triangle, which means that we cannot use the theorem with this triangle.” Back Go to Evaluation 1

How do you know which side of the triangle is the hypotenuse? “The hypotenuse is the side opposite the right angle in a right triangle. So, in the triangle below, side c would be the hypotenuse.” Back Go to Evaluation 1

Evaluation: Quiz: Question #1 1. In the triangle below, what is the measure of the missing side? a) 2 2 b) 7 7 c) d) ? Hint

Hint → Quiz: Question #1 “For the triangle below, the Pythagorean Theorem states a 2 + b 2 = c 2. So, c = √(a 2 + b 2 ).” Back 1

“Alrighty! You got it right! This next one should be easy then.” NextBack 1

“Oh, not quite. Remember, the formula is a 2 + b 2 = c 2, where c is the hypotenuse.” Back 1

Evaluation: Quiz: Question #2 1. What is the perimeter of the triangle below? a) b) 7 7 c) 2 2 d) Hint

Hint → Quiz: Question #2 “For perimeter of the triangle below is a + b + c. Use the Pythagorean Theorem to find the missing side, c, and then compute the perimeter.” Back 1

“Correct! You got it right! Let’s see if you can get the last one.” NextBack 1

“Oh, you got it wrong but try again. Remember, to find the perimeter, you need to add up all the sides of the triangle. So start by finding the missing side.” Back 1

Evaluation: Quiz: Question #3 1. What is the area of the triangle below? a) b) 9 9 c) 6 6 d) Hint

Hint → Quiz: Question #3 “The area of a triangle is the base times the height divided by 2. so, in the triangle below, the base is a and the height is b. Therefore, the area is (a x b)/2.” Back 1

“Yay! That’s entirely right! That was the last quiz question so fantastic job.” NextBack 1

“Oh, so sorry. Remember, for a triangle, the area is equal to the base times the height divided by 2. So, the area of the triangle below is (a x b)/2 Back Area = (a x b)/2 1

“If you want to review what we did today, select one of the options below. Otherwise, click next.” Back to the Beginning Review Pythagorean Theorem Retake Quiz Next 1

“Good job today! Looks like you’ve mastered the Pythagorean Theorem. For more practice, visit this website: theorem-game.php Works Cited Back 1

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