The Crooked Foundation The Bird House. 100ft. Inter-Island Journey. East Fence. 150ft. South Fence

Transcription

1 Opening Per Date It is another beautiful day on the Big Island, and Grandma is out and about planning her net set of projects. First she wants to build a bird house for her new team of homing pigeons. The last one that her grandson helped make had an etremely crooked foundation and she wants to make sure that this new one is a perfect rectangle. Second, a gravel path from her house to the birdhouse will need to be constructed so that she can feed the birds without walking through the wet grass. Finally, since she has these homing pigeons, she wants to see if everything she has heard about them is true. Grandma has several friends on different islands and has read that these pigeons can fly up to 1000 miles without stopping. It would be a fun hobby to be able to send messages to her off-island friends. But how far is it as the bird flies to each island? She will have to investigate this further. The Crooked Foundation The Bird House Draw a right triangle with the longest side connecting the house and the birdhouse. The quadrilateral is a parallelogram. Label all congruent segments. C B 100ft B D A E Inter-Island Journey 150ft House A East Fence Draw a right triangle with the longest side connecting points A and B. B A South Fence Page 1

2 13.2 The Pythagorean Theorem Per Date Grandma notices that all of these projects involve right triangles in some way. So she cracks open the old book and does a little research to make sure that her math skills are still up to the task. Investigation 1 - Find the areas of each of the squares adjacent to the sides of the triangle by counting the blocks within them. Then write a conjecture about a possible relationship between these areas. Eample 1!! =!! =!! = c a c a b Eample 2!! =!! =!! = b a a c b b c The area of is equal to the sum of the areas of Page 2

3 13.2 The Pythagorean Theorem Per Date Investigation 2 1. Label the three sides of the right triangle below with a, b and c. Side c should be the longest, opposite the right angle. 2. Using the graph paper provided on the following page, cut out two squares with side length a, and two squares with side length b. 3. Glue one of each to the appropriate sides of the triangle. 4. Now use the remaining two squares (a squared and b squared) to create a third square on side c. (hint: you may need to cut up the squares). 5. What is the area of the square adjacent to side a? How about b? 6. What is the area of the square adjacent to side c? 7. What is the length side c? 8. Write a formula for this relationship between side a, b and c. Page 3

4 13.2 The Pythagorean Theorem Per Date Cut out and use for Investigation 2. Page 4

5 13.3 Pythagorean Theorem Proof Per Date The Pythagorean Theorem (#THM): In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.!! leg a c b leg!! *Hypotenuse!! +!! =!! *The Hypotenuse (#VOC) is the longest side of the right triangle. It is always opposite the right angle.!! Practice: Find the area () of the missing square. Then find the side length of square.! 25!! 34!! Area of =!! Side Length =! Page 5

6 13.3 Pythagorean Theorem Proof Per Date To prove this relationship algebraically, we first begin with a right triangle. Then we draw an altitude from the hypotenuse to the right angle. a c This creates a similar triangle (the small one with side ) within the larger one. They are similar because they are both right triangles and share a verte. This makes them similar by the AA similarity postulate. b c a a c - The Pythagorean Theorem Given: A right triangle with legs a and b and hypotenuse c. Prove:!! +!! =!! This proof can be done purely algebraically. To begin, use the fact that the two triangles drawn are similar to write two different proportions, each containing. Proportion 1 Proportion 2 b Page 6

7 13.3 Pythagorean Theorem Proof Per Date Now, use algebra to rearrange each proportion to isolate the s. Then, use substitution to eliminate the. Finally, use algebra to simplify and rearrange the equation to produce!! =!! +!!. Complete the proof in the space below. 1.!! =!! Statements Reasons 1. Proportion !!!! =!! Solve for. 3. Proportion Solve for This property also goes in the other direction. The Converse of the Pythagorean Theorem (#THM): If the sum of the squares of the lengths of the two sides of a triangle is equal to the square of the length of the longest side, the triangle is a right triangle. B EXAMPLE!! +!! =!! 12 m 20 m 12! + 16! = 20! = = 400 C 16 m A The equation is true therefore!!"# = 90 This property is useful for determining if a triangle contains a right angle as shown in the eample above. The property will be eplored more fully in the net lesson. Page 7

8 13.4 Homework 1 Per Date 1. Which sides of the triangle below are legs? Which is the hypotenuse? Describe how you made your decision. L K J 2. Find the area in the figure below. Then find the side length of square. 12!! 13!!! Area of =!! Side Length =! 3. Find the area in the figure below. The find the side length of square.! 18!! 23!! Area of =!! Side Length =! Page 8

9 13.5 Reviewing Square Roots Per Date Square Roots Review For each epression, write an equivalent epression that has no perfect squares under the radical and contains only one term. Recall the property that allows us to split up radicals into their factors and then simplify any perfect squares:!! =!! Eample:!" =!"! =!"! =!! Practice Write each epression with no perfect squares under the radical. Then use a calculator to approimate its value to the thousandths place. a. 75 b. 117 c d. 512 You can also use these properties to help solve equations. Solve each equation for and write your answer in the form described above. Remember, quadratic equations always have two solutions. E.!! =!"!! =!"! = ±!"! = ±!! a.!! = 20 b.! = ± 3! + 4! c.! = ± 25! 3! Page 9

10 13.6 Missing Sides Per Date The Crooked Foundation Grandma is starting to build her birdhouse and wants to make sure that the frame is perfectly square. She has noticed that just using a carpenter s square is not adequate because wood is often warped. She needs all 4 corners to be perfectly square. Luckily, you remember something from math class. The diagonals of a rectangle are congruent Nice! So if we can make the diagonals the same length we should be able to make a perfect rectangle. So, what we need to do is calculate eactly how long the diagonal needs to be so we can measure it eactly, and move the sides to form that distance. Crooked 18in 24in Not - Crooked 18in 24in Page 10

11 13.6 Missing Sides Per Date You can break out a right triangle from the foundation above and use the Pythagorean theorem to find the missing hypotenuse. Find the length by filling in the blanks. EXAMPLE!! +!! =!!! +! =!! + =!! c a 18in =!! ± =! ± =! b 24in a. The side length of 24in turned out to be a little too long. Grandma decided to keep the same diagonal, but shortened the length to 20in. Use the hypotenuse you calculated in the first problem to find the new width (). Write an equation by filling in the known values into the Pythagorean theorem. Then solve for. ( )! + 20! = ( )! 20in in Page 11

12 13.6 Missing Sides Per Date b. This also seems too long. Grandma tries a whole new different strategy. This time she is going with a square foundation, but with the same diagonal length as before. Find the new side length, and round your answer to the hundredths place. in That last bird house was only a miniature scale of what Grandma really wants. She wants to go big, and draws out plans for a giant delue bird house, 30ft by 24 ft. Her grandson has also started construction and has made the diagonal 40 ft. 1. Is this the correct diagonal length to make the frame square? 2. If not, what would be the correct length? 40ft 30ft 24ft ft Page 12

13 13.6 Missing Sides Per Date Practice: Find the missing side of each figure using the Pythagorean theorem. Write your answer in radical form, with no perfect squares under the radical, and approimate your answer to the nearest thousandths place. 1. L 5 in K 12 in J 2. O 3 5 m N M 3. B 15 in C A Page 13

14 13.6 Missing Sides Per Date Find the missing length in the shapes below ft 5. C 15m B 5m D 8m A Page 14

15 13.6 Missing Sides Per Date Investigation 3 - The Pythagorean Theorem directly relates to right triangles, but it can also be used to help identify other types of triangles. Use your compass (or physical object such as a straw or spaghetti) to construct a triangle with the side lengths provided below on a separate paper. Then use your protractor to measure the interior angles. Record the type of triangle in the bo (acute, obtuse right). Triangle 1 a = 3in Δ1 b = 4in c = 5in Triangle 2 a = 3in Δ2 b = 4in c = 5.5 in Triangle 3 a = 3in Δ3 b = 4in c = 4.5 in a. Not all the triangles above are right triangles; however, using the Pythagorean theorem on them anyway might still tell us something. Plug in the values for legs a and b in each of the triangles above and solve for c on the same paper you drew the triangles on. Then compare the resulting length of c to the actual measurement of the triangle. Page 15

16 13.6 Missing Sides Per Date b. When the Pythagorean theorem gives you the same length for c, you should have a right triangle by the Converse of the Pythagorean Theorem. Fill out the table below to discover which triangle contains a right angle. Triangle!!!!!! Right Triangle? Triangle 1 Triangle 2 Triangle 3 c. Make a conjecture: If the sum of the areas of the two squares on the legs of a triangle is greater than the area of the square on the hypotenuse then the triangle will be. d. Make a conjecture: If the sum of the areas of the two squares on the legs of a triangle is less than the area of the square on the hypotenuse then the triangle will be. Practice: Use the converse of the Pythagorean Theorem to determine if each of the triangles is a right triangle. If it is not, indicate if it is obtuse or acute. 1. A triangle with side lengths a = 8m, b = 6m and c = 10m. 2. A triangle with side lengths! in,!" in,!" in. 3. A triangle with side lengths 3in, 7in, 5in. Page 16

17 13.7 Homework 2 Per Date 1. Find the missing length of each triangle using the Pythagorean theorem. Write your answer in simplest radical form and then approimated to the thousandths place. a. b. 18m 9m 15m 8m! =!! =! 2. Find side length AD in the figure below. 16m 18m 7m! =! Page 17

18 13.8 The Distance Formula Per Date The Bird House - Grandma now turns her attention to the birdhouse for her new carrier pigeons. Her net step is to construct a straight gravel path from her house to the birdhouse. She wants to know how long that straight path will need to be so that she can purchase the correct amount of gravel. The birdhouse will be in the north-east corner of her rectangular mowed yard, while the corner of her house currently stands 45ft from the west fence and 22ft from the south fence. a. Draw a right triangle with the hypotenuse connecting the corner of the House and the birdhouse. Label the distances of the legs of the triangle. b. Use the Pythagorean theorem to find the length of the path. Draw the right triangle and show your work in the space below. 100ft B 150ft East Fence House A South Fence Distance: This is interesting. Is there a way to indicate the position of the house and the birdhouse using a pair of numbers? If the origin were at the Southwest corner, list the coordinates of the house and birdhouse. House Coordinates (, ) Bird House Coordinates (, ) Page 18

19 13.8 The Distance Formula Per Date Inter-Island Journey Investigation 4 - Finally, since Grandma has all these homing pigeons, she wants to see if everything she has heard about them is true. She has friends on several different islands and has read that these pigeons can fly up to 1000 miles without stopping. It would be fun to be able to send messages to her off-island friends. But how far would it be as the bird flies to each island? She decides to investigate. Time to pull out the old map. Map scale 1 unit = 15 mi 1. Draw a right triangle with the hypotenuse connecting Hilo (A) and Honolulu (C), and the legs following the grid lines. 2. Use the Pythagorean theorem to find the relative length of the hypotenuse. Then find the actual distance using the map scale. Page 19

20 13.8 The Distance Formula Per Date There has got to be an easier way than drawing on an old paper map. What if we didn t have the map, only the coordinates? Start with a simple eample. Hilo (A): (11, -2) Honolulu (B): (-2, 6) We will still need to use the formula!! +!! =!! but maybe we can tweak it a bit. Fill in the blanks below to help generate the Distance Formula. The lengths of the legs can be found by finding the difference between the and y coordinates of the start and end points. 3. Distance between the values:. This is the leg on the triangle. It is units long. 4. Distance between the y values:. This is the leg on the triangle. It is units long. 5. The order does not matter because of the even eponent. Rewrite the formula by replacing a with the difference you found in part 3. ( )! +!! =!! 6. Now replace b with the difference from part 4. ( )! + ( )! =!! 7. Finally, solve for the distance, c by taking the positive square root of both sides. ( )! + ( )! =! 6. Approimate the epression above. Did you get the same distance as you did in question 2? 7. Now, rewrite this equation in a more general form. Replace the numbers from part 7 with s and y s. Because there are two points, we will call them (!!,!! ) and (!!,!! ). The Distance Formula (#THM):! = Page 20

21 13.8 The Distance Formula Per Date Practice 3.3 (Use the map provided in Investigation 4.) 1. Find the distance from Hilo (A) to Koloa (D) without drawing a triangle in the coordinate grid. Use the distance formula. 2. Check your work on the map by drawing the triangle and measuring. 3. Grandma s friend in Kahului (B) has a daughter who is out in the ocean fishing. Her coordinates are (10, 22). How far is the boat from Kahului? 4. Grandma wants to send her pigeon from Hilo (A) to her grandson s sailboat which is out at coordinates (20, -55). Will the pigeon be able to make it to the boat? Eplain. Remember, it has a maimum range of 1000 mi. Page 21

22 13.8 The Distance Formula Per Date 5. It s been a windy day and the grandson s boat has drifted further out to sea. It is now at (20, -72). Will the pigeon still be able to make it to the sailboat? Eplain. 6. Place a dot on your current approimate location on the map of the Hawaiian Islands. Then choose one of the four starting cities on a different island than your own. Find the distance between those two points. 7. What would happen if the or y components of the two points had the same numerical value? How would that be reflected in the formula? Give an eample. Page 22

23 13.9 Homework 3 Per Date A map of Alaska is shown below. Map Scale 1 unit = 80 mi a. Find the distance between points A and B. Remember to use the scale. b. Find the distance between points C and D. c. A small island in the pacific is located at coordinate E(44, -68). Find the distance from point B to the island. Page 23