How can we organize our data? What other combinations can we make? What do we expect will happen? CPM Materials modified by Mr.
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1 Common Core Standard: 8.G.6, 8.G.7 How can we organize our data? What other combinations can we make? What do we expect will happen? CPM Materials modified by Mr. Deyo
2 Title: IM8 Ch What Is Special About A Right Triangle? Date: Learning Target By the end of the period, I will identify the relationship between side lengths of a right triangle as the Pythagorean Theorem and apply that relationship to solve problems. I will demonstrate this by completing Four Square notes and by solving problems in a pair/group activity.
3 Home Work: Sec Desc. Date Due Review & Preview 3 Problems 9 75, 9 78, 9 79
4
5 Vocabulary 1) Right Triangle 2) Leg(s) 3) Hypotenuse 4) Pythagorean Theorem
6
7 9.2.2 What Is Special About A Right Triangle? In Lesson 9.2.1, you saw that for three lengths to form a triangle, they must be related to each other in a special way. Today, you will investigate a special relationship between the side lengths of right triangles. This relationship will allow you to find the length of a missing side. 9 68a,b Use your patterns from Lesson to decide if the squares listed below will form a right triangle. a) Squares with side lengths 6, 8, and 10 meters b) Squares with areas 64 in 2, 100 in 2, 144 in 2
8 9 68c,d Use your patterns from Lesson to decide if the squares listed below will form a right triangle. c) Two squares with side length 5 feet and a square with area 50 square feet. d) Explain how you know whether three squares will join at their corners to form a right triangle.
9 9 69a c. THE PYTHAGOREAN RELATIONSHIP Based on your work so far, if you know the area of three squares, you can tell if they will connect at their corners to form a right triangle. But what if you know that a triangle has a right angle? Will the lengths of the sides be related in this way? Work with your team to look more closely at side lengths of some right triangles. a) On centimeter graph paper, form a right angle by drawing one 5 cm length and one 12 cm length as shown at right. If you do not have centimeter graph paper, then use any graph paper to draw and measure these lengths with a ruler. After drawing the two lengths, create a right triangle by connecting the ends of the two lengths with a third side. b) With a ruler, measure the longest side of the triangle in centimeters and label this length. If you do not have a centimeter ruler or you are using another kind of graph paper, create your triangle using 5 and 7 grid units. Then use an edge of the page and the grid lines as your ruler. c) Visualize a square connected to each side of the right triangle in part (b). On your paper, sketch a picture like the one at right. What is the area of each square? Is the area of the square that is connected to the longest side equal to the sum of the areas of the other two squares?
10 9 69a c Work Area
11 9 69d,e. THE PYTHAGOREAN RELATIONSHIP Based on your work so far, if you know the area of three squares, you can tell if they will connect at their corners to form a right triangle. But what if you know that a triangle has a right angle? Will the lengths of the sides be related in this way? Work with your team to look more closely at side lengths of some right triangles. d) Check this pattern with a new example. Draw a new right angle on the centimeter paper like you did in part (a). This time, use 9 cm and 12 cm lengths. Connect the endpoints to create a triangle, and measure the third side. Create a sketch for this triangle like the one you created in part (c), and find the areas of the squares. Is the area of the square that is connected to the longest side equal to the sum of the other two areas? e) The two shortest sides of a right triangle are called the legs, and the longest side is called the hypotenuse. You previously wrote a statement about the relationship between the areas of squares drawn on the sides of a right triangle. Now use words to describe the relationship between the lengths of the legs and the length of the hypotenuse. The length of squared plus the length of the squared is equal to the squared.
12 9 69d e Work Area
13 9 70a. The relationship you described in part (e) of problem 9 69 is called the Pythagorean Theorem. It states that in a right triangle, the length of one leg squared plus the length of the other leg squared is equal to the length of the hypotenuse squared. It can be written as an equation like this: (leg A) 2 + (leg B) 2 = (hypotenuse) 2 a) Use the Pythagorean Theorem to write an equation for the diagram below. Then find each missing area.
14 9 70b,c. Use the Pythagorean Theorem to write an equation for the diagram below. Then find each missing area. b) c)
15 9 71a,b. In Lesson you found a relationship between the squares of the sides of triangle and the type of triangle (acute, obtuse, or right). You discovered that if the sum of the squares of the two shortest sides in a triangle equals the square of the length of the longest side, then the triangle is a right triangle. Use this idea to determine whether the lengths listed below form a right triangle. Explain your reasoning. a) 15 ft., 36 ft., and 39 ft. b) 20 in., 21 in., and 29 in.
16 9 71c,d. In Lesson you found a relationship between the squares of the sides of triangle and the type of triangle (acute, obtuse, or right). You discovered that if the sum of the squares of the two shortest sides in a triangle equals the square of the length of the longest side, then the triangle is a right triangle. Use this idea to determine whether the lengths listed below form a right triangle. Explain your reasoning. c) 8 yd., 9 yd., and 12 yd. d) 4 m, 7 m, and 8 m
17 9 72a,b. Find the area of the square in each picture. a) b)
18 9 72c,d. Find the area of the square in each picture. c) d)
19 9 73. How long is the missing side of each triangle in parts (b) and (c) of problem 9 72? Be prepared to explain your reasoning. b) c)
20 9 74. If you have 24 square tiles, how many different rectangles can you make? Each rectangle must use all of the tiles and have no holes or gaps. Sketch each rectangle on graph paper and label its length and width. chapter/ch Can you make a square with 24 tiles?
21 9 75. Lydia has four straws of different lengths, and she is trying to form a right triangle. The lengths are 8, 9, 15, and 17 units. Which three lengths should she use? Justify your answer. chapter/ch
22 9 76. The Wild West Frontier Park now offers an unlimited day pass. For $29.00, visitors can go on as many rides as they want. The original plan charged visitors $8.75 to enter the park, plus $2.25 for each ride. Write an equation to determine the number of rides that would make the total cost equal for the two plans. Solve the equation. chapter/c
23 9 77 Complete the following table. chapter/ch9/lesson/9.2.2/pr Try graphing the points you have. a) Write the rule for the table. y = ( )x + ( ) b) What is the slope? m = c) What is the y intercept? ( 0, )
24 9 78a. Solve for x. chapter/ch9/lesson/9.2.2/pr If m 1 = 3x 18 and m 5 = 2x + 12, find x.
25 9 78b. Solve for x. chapter/ch9/lesson/9.2.2/pr If m 3 = 4x 27 and m 6 = 1x + 39, find x.
26 9 78c. Solve for x. chapter/ch9/lesson/9.2.2/pr If m 4 = 49 and m 6 = 5x + 41, find x.
27 9 79. Calculate the value of x. chapter/ch9/lesson/9.2 a) b)
IM 8 Ch Does It Always Work. Common Core Standard: Is the triangle a right triangle? Who is Pythagoras? CPM Materials modified by Mr.
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