CCM8 Unit 7: Pythagorean Theorem Vocabulary

Transcription

1 CCM8 Unit 7: Pythagorean Theorem Vocabulary Base Exponent Hypotenuse Legs Perfect Square Pythagorean Theorem When a number is raised to a power, the number that is used as a factor The number that indicates how many times the base is used as a factor The longest side of a right triangle, opposite the right angle The two sides of a right triangle that make up the right angle The square of a rational number Theorem that states in a right triangle the length of the hypotenuse (c) squared is equal to the sum of the squares of the lengths of the legs a 2 + b 2 = c 2 Pythagorean Triple Right Angle Right Triangle Square Root Three positive integers that make up the lengths of the sides of a right triangle Angle that measures 90 degrees Triangle with one right angle One of the two equal factors of a number 2017/2018 pg 1

2 Pythagorean Theorem Investigation The Pythagorean Theorem allows us to find the third side length of a right triangle if we know two of the triangle s side lengths. Did you know that sides of a right triangle have special names? The two sides that form the right angle are called the legs and are noted as sides a and b. The side across from the right angle is always the longest side of the right triangle. This side is called the hypotenuse and is noted as side c. Part One: Identify the legs and the hypotenuse of the right triangle below and label the side lengths of the legs of the triangle. Leg a is units in length. Leg b is units in length. 1. On the grid below create a square from each of the legs, so that each square shares a side length with a leg of the right triangle. 2. Find the area of each of your new squares then find the sum of these square areas. The square of side a is square units. The square of side b is square units. The sum of those two sides squared = 3. Now compare the combined area of these two squares to side c s square. Side c s square has been copied, rotated and lined up on the grid lines below this diagram for easy comparison. 4. The square to the left is the identical size square as the square of the hypotenuse. How do the areas of the side a s and b s squares compare to the area of side c s square? 2017/2018 pg 2

3 Part Two: 1. Repeat your process from the last problem and record your findings. We have included a rotated copy of the square of side c so that you can easily find its area and compare it to the sum of the areas of sides a and b. What do you notice about the relationship between sides a, b and c? 2. What relationship do you notice between the legs and hypotenuse of this right triangle? How does this relate to what you observed in part one? Part Three: 1. The Pythagorean theorem can be used to find the length of one side of a right triangle if you know the other two sides of that triangle. The Pythagorean theorem is stated as. Explain how the diagrams from part one and two demonstrate the Pythagorean theorem. 2. In parts one and two you were given the leg lengths of the triangles and asked to find the length of the hypotenuse. How might you use the Pythagorean theorem to find a leg length when given the length of the triangle s hypotenuse and one leg? If it helps, consider using an example or making a diagram in your explanation. Yummymath.com 2017/2018 pg 3

4 NOTES: Pythagorean Theorem Deals with the relationship between the side lengths in a RIGHT TRIANGLE Sides A and B are both called Side C is called the This side is always the and it is opposite from the right angle. The relationship that is true for every right triangle, as stated in the Pythagorean Theorem, is: Examples: Calculate the missing sides of a right triangle. Examples: Determine whether a triangle with the given side lengths is a right triangle. **Pythagorean Triple: a) 7 in, 24 in, 25 in b) 6 m, 7 m, 85 m c) 8 cm, 10 cm, 12 cm 2017/2018 pg 4

5 Examples: Application of the Pythagorean Theorem 1. A 24 foot wire is attached to an electrical pole. If the pole is 20 feet tall, how far is the wire sitting from the base of the pole? Round answers to the nearest tenth of a foot. 2. The new flat-screen television your parents bought has a length of 36 inches and a width of 19.6 inches. To the nearest inch, what is the length of the television s diagonal? 3. If the right triangle shown is an isosceles triangle with a hypotenuse that measures 18 cm, what would be the length of each leg be to the nearest tenth of a cm? 4. Find the perimeter of a right triangle is one leg is 5.6 ft and the area is 33.6 ft /2018 pg 5

6 Find the length of the third side of the right triangle. Give an exact answer and an approximation to three decimal places c b b a In a right triangle, find the length of the side not given. Give an exact answer and an approximation to three decimal places. 5. a 15, c b 12, c 18 7) If one leg of a right triangle is 12 and the other leg is 16, what is the length of the hypotenuse in this right triangle? 8) Find the missing measure if a and b are the legs of the right triangle and c is the hypotenuse, with a = 11 and c = /2018 pg 6

7 Distance formula and Applications: Make the line into the hypotenuse of a right triangle and use the Pythagorean Theorem ) Find the missing side of the triangle and find the area and perimeter of the triangle. 5) A telephone pole support cable attaches to the pole 20 feet high. If the cable is 25 feet long, how far from the bottom of the pole does the cable attach to the ground? 2017/2018 pg 7

8 Homework The distance from each consecutive base on a Major League baseball diamond is 90 feet and it can be necessary to determine how far the catcher will have to throw to get the ball from home plate to 2 nd base or the distance of the throw from the 3 rd baseman to 1 st base. We will use the Pythagorean Theorem to find answers to some of these questions. A. What other shape is a baseball diamond? B. Explain how you might use a right triangle and the Pythagorean Theorem to determine how far a catcher has to throw from home plate to 2 nd base. C. Now, set up the problem so that you can use the Pythagorean Theorem to find out how far the catcher will have to throw the ball from home plate to 2 nd base to the nearest foot. What is the distance measured from home plate to 2 nd base in feet and inches? D. How far will the third baseman have to throw a baseball from 3 rd base to 1 st base? How can you determine this answer from the work that you have already done. Explain. 5. A wire 30 feet long is stretched from the top of a flagpole to the ground at a point 15 feet from the base of the pole. How high is the flagpole? Round to nearest tenth. 6. George wants to put a fence around half of his rectangular flower garden. He is will cut the garden in half diagonally. If one side of the garden is 10 feet and the diagonal is 18 feet, about how much fencing will George need. 2017/2018 pg 8

9 Unit Review: Find the missing measure of each right triangle. Round to the nearest tenth if necessary. 2017/2018 pg 9

10 Determine whether triangle with the given sides is a right triangle , 44, , 10, Find the perimeter if on leg of a right triangle is 12 m and the area is 60 m 2 4. What is the length of a diagonal of a rectangular picture whose sides are 12 inches by 17 inches? Round to nearest tenth. 5. Ross wants to the area of his vegetable garden to know how many seeds to buy. He measures one side of the garden as 24 feet and the diagonal as 25 feet. What is the area of Ross garden? 6. Troy drove 8 miles due east and then 5 miles due north. How far is Troy from his starting point? Round to nearest tenth. 7. What is the perimeter of a right triangle if the hypotenuse is 15 centimeters and one of the legs is 9 centimeters? 2017/2018 pg 10

11 8. A 20-foot ladder leaning against a wall is used to reach a window that is 17 feet above the ground. How far from the wall is the bottom of the ladder? Round to nearest tenth. 9. Tina measures the distances between three cities on a map. The distances between the three cities are 45 miles, 56 miles, and 72 miles. Do the positions of the three cities form a right triangle? 10. What is the area of a triangle if the hypotenuse is 13 meters and one of the legs is 5 meters? 11. A ladder 17 feet long is leaning against a wall. The bottom of the ladder is 8 feet from the base of the wall. How far up the wall is the top of the ladder? 2017/2018 pg 11