CHAPTER Right Triangles Hiking is the most popular outdoor activity in the United States, with almost 40% of Americans hiking every year. Hikers should track their location and movements on a map so they don t get lost in wilderness areas. You will use the Pythagorean Theorem to calculate the distance hikers have travelled..1 Get Radical or (Be) 2! Radicals and the Pythagorean Theorem p. 17.2 The Pythagorean Theorem Disguised as the Distance Formula! The Distance Formula and Midpoint Formula p. 181. Drafting Equipment Properties of 45º 45º 90º Triangles p. 189.4 Finishing Concrete Properties of 0º 60º 90º Triangles p. 19 Chapter Right Triangles 171
172 Chapter Right Triangles
.1 Get Radical or (Be) 2! Radicals and the Pythagorean Theorem Objectives In this lesson, you will: Simplify radical expressions. Use the Pythagorean Theorem. Key Terms radical expression radicand rationalizing the denominator Problem 1 Simplifying Radicals To square a number, multiply the number by itself. Take the square root of that value to undo the process. For instance, if 12 2 144, then 144 12. A radical expression is an expression that involves a radical symbol. The radicand is the expression enclosed by the radical symbol. In the radical expression 144, 144 is the radicand. To simplify a radical expression, write the radicand as a product of the largest perfect square factor and another factor. Then simplify. 1. Can 20 be simplified? Explain. 2. Can 21 be simplified? Explain.. Can 64 be simplified? Explain. Lesson.1 Radicals and the Pythagorean Theorem 17
4. Simplify each radical expression completely. a. 18 b. 5 24 c. x 5, for x 0 d. 18x, for x 0 e. 12x y 5, for x 0, y 0 7 5. Can be simplified? Explain. 4 7 6. Can be simplified? Explain. 2 The process of eliminating a radical from the denominator is called rationalizing the denominator. To rationalize the denominator, multiply by a form of one so that the radicand of the radical in the denominator is a perfect square. 7. Simplify each radical expression completely. 5 a. 174 Chapter Right Triangles
b. 2 5 c. 2 8 8. Describe what it means to simplify a radical expression. 9. Luke claims that 49 7 because 7 2 49. Macey does not agree with Luke. She insists that 49 7 because ( 7) 2 49. Who is correct? Explain your reasoning. Lesson.1 Radicals and the Pythagorean Theorem 175
Problem 2 Moving Furniture Recall that the Pythagorean Theorem states that a 2 b 2 c 2 where a and b represent the length of the legs of a right triangle and c represents the length of the right triangle s hypotenuse. 1. An entertainment center is carried into your house on its side to fit through the front door and is placed on the floor on its side. The entertainment center is 7 feet high and 5 feet long. Will it hit the 8.5-foot ceiling when it is tilted upright? Justify your conclusion. 8.5 ft 7 ft 5 ft Ceiling Floor Problem Skis 1. The interior of a vehicle has a width of 40 inches and a length of 62 inches. You would like to lay a 6.5-foot pair of skis along the bottom of the interior. Would the skis fit? 176 Chapter Right Triangles
Problem 4 Hiking You are backpacking for the weekend with friends. You park the car and hike into the woods walking 2 miles south, 1 mile east, miles south, and 4 miles east before setting up camp. Sam thinks the campsite is 10 miles from the car, and Gwen thinks the campsite is 7.24 miles from the car. 1. How did Sam calculate his estimate of 10 miles? 2. Why is Sam s calculation incorrect?. How did Gwen calculate her estimate of 7.24 miles? 4. Why is Gwen s calculation incorrect? 5. How far is the campsite from the car? How do you know? 6. Would Gwen s method work if the distances walked were different? If so, provide an example. Lesson.1 Radicals and the Pythagorean Theorem 177
Problem 5 Consider the diagram shown. Each of the four small circles has a radius of 4 inches and is tangent to each of the other circles and to the large circle. Connect the centers of the small circles. 1. What is the length of the radius of the large circle? 178 Chapter Right Triangles
Problem 6 A Riddle 1. Calculate the depth of the water in a ground well using the following clues. Clue 1: You place a stick vertically into the well, resting against the inner well wall and perpendicular to ground as drawn. The stick touches the bottom of the well. An 8-inch portion of the stick is above the surface of the water. 8 in. Clue 2: Without moving the bottom of the stick, you rest the top of the stick against the opposite wall of the well. The top of the stick is even with the surface of the water. 6 in. Clue : The diameter of the well is 6 inches. Lesson.1 Radicals and the Pythagorean Theorem 179
Problem 7 Exact vs. Approximate A right triangle has legs measuring 4 inches and 8 inches. 1. What is the exact length of the hypotenuse? Write your answer as a fully simplified radical expression. 2. What is the approximate length of the hypotenuse? Write your answer as a decimal.. When does it make more sense to use the exact answer for the hypotenuse? 4. When does it make more sense to use the approximate answer for the hypotenuse? Be prepared to share your solutions and methods. 180 Chapter Right Triangles
.2 The Pythagorean Theorem Disguised as the Distance Formula! The Distance Formula and Midpoint Formula Objectives In this lesson, you will: Derive and use the distance formula. Use the midpoint formula. Key Terms distance formula midpoint midpoint formula Problem 1 1. Plot the points (1, 2) and (, 7) on the grid. 2. Connect the points with a line segment. Draw a right triangle with this line segment as the hypotenuse.. What are the lengths of each leg of the right triangle? Lesson.2 The Distance Formula and Midpoint Formula 181
4. Use the Pythagorean Theorem to calculate the length of the hypotenuse. Round your answer to the nearest tenth. 5. What is the distance between the points (1, 2) and (, 7)? Problem 2 1. Plot any two points on the grid. Label the points (x 1, y 1 ) and (x 2, y 2 ). 2. Connect the points with a line segment. Draw a right triangle with this line segment as the hypotenuse.. What are the lengths of each leg of the right triangle in terms of x 1, x 2, y 1, and y 2? 4. Use the Pythagorean Theorem to calculate the length of the hypotenuse. 182 Chapter Right Triangles
5. What is the distance, d, between the points (x 1, y 1 ) and (x 2, y 2 )? Problem 1. Use the distance formula derived in Problem 2 Question 5 to calculate the distance between the points (1, 2) and (, 7). 2. Compare your answer from Problem Question 1 to your answer from Problem 1 Question 5. What do you notice?. To determine the distance between two points, is it easier to use the Pythagorean Theorem or the distance formula? Explain. Lesson.2 The Distance Formula and Midpoint Formula 18
Problem 4 1. Use the distance formula to calculate the distance between each pair of points. a. ( 6, 4) and (2, 8) b. ( 5, 2) and ( 6, 10) c. ( 1, 2) and (, 7) 2. The distance between the points (, 0) and (4, y) is 65. Solve for y. 184 Chapter Right Triangles
Problem 5 In addition to calculating the distance between two points, sometimes it is important to locate the point halfway between two points. The midpoint of a line segment is the point on the segment that is equidistant from the endpoints of the segment. The midpoint of a segment bisects the segment. 1. Locate the values and 7 on the number line. x 2. What value on the number line is equidistant from and 7? Explain how you determined the answer to this question.. Consider the points x1 and x2 on a number line, with x2 to the right of x1. Determine a simplified expression for the midpoint of the segment with endpoints at x1 and x2. Lesson.2 The Distance Formula and Midpoint Formula 185
The process that you used to calculate a midpoint along the number line can be extended to calculate a midpoint in the coordinate plane. The midpoint formula states that the midpoint between any two points (x 1, y 1 ) and (x 2, y 2 ) is ( x 1 x 2, y 1 y 2. 2 2 ) 4. Explain how the midpoint formula is similar to the process of calculating a mean? 5. Use the Midpoint Formula to determine the midpoint of a line segment with the given endpoints. a. (0, 5) and (4, ) b. ( 10, 7) and ( 4, 7) c. (, 1) and (9, 7) 6. The point (7, 1) is the midpoint of a line segment with endpoints (8, 2) and (x, 0). Solve for x. 186 Chapter Right Triangles
Problem 6 1. Graph the points (2, 12), ( 2, 4), (4, 0), and (8, 8) on the grid. 2. Connect the four points to form a quadrilateral. Draw the diagonals of the quadrilateral.. Calculate the midpoint of each diagonal. 4. What do you notice about the midpoints of the diagonals? 5. What type of quadrilateral is defined by the points? How do you know? Lesson.2 The Distance Formula and Midpoint Formula 187
Problem 7 1. Sketch a square on the grid. Label the coordinates of the vertices. 2. Determine the midpoints of each side of the square. Label the coordinates of each midpoint.. Connect the midpoints. What figure is formed? How do you know? 4. What is the area of the original square? 5. What is the area of the figure formed by the midpoints of the square? 6. What do you notice about these two areas? Be prepared to share your solutions and methods. 188 Chapter Right Triangles
. Drafting Equipment Properties of 45 o 45 o 90 o Triangles Objectives In this lesson, you will: Find unknown side lengths of 45º 45º 90º triangles. Find areas of 45º 45º 90º triangles. Key Term 45º 45º 90º triangle Problem 1 Drawing a Triangle Architects and engineers must be skilled at making technical drawings as part of their job. Technical drawings are precise models of objects, buildings, or parts that indicate exact shapes and give precise measurements. Most of today s technical drawings are done on the computer using a CAD (computer-aided design) program. Drawings can be done by hand, and one of the instruments used to create hand drawings is a right triangle. Take Note An isosceles right triangle is a triangle with a right angle and two equal-length legs. One of the right triangles used to create a technical drawing is an isosceles right triangle. A. Use the following grid to create an isosceles right triangle. B. Identify the lengths of the legs of your triangle. Label these lengths on your diagram in part (A). Then label the length of the hypotenuse as c. C. Complete the following equation, which relates the lengths of the sides of the triangle. 2 2 2 Now simplify the right-hand side of the equation. 2 2( 2 ) Lesson. Properties of 45º 45º 90º Triangles 189
Finally, complete the following statements to write the length of the hypotenuse as a radical in simplest form. c c c D. Draw a different isosceles right triangle on the grid in part (A). Label the legs and hypotenuse of the triangle like you did in part (B). E. Write a radical expression in simplest form for the length of the hypotenuse of the right triangle in part (D). F. What can you conclude about the lengths of the legs when compared to the hypotenuse in an isosceles right triangle? Investigate Problem 1 1. Measure the angles in your triangles on the grid in part (A). What do you notice? What can you conclude about the angles that are not right angles in an isosceles right triangle? 2. Just the Math: 45º 45º 90º Triangle An isosceles right triangle is often called a 45º 45º 90º triangle. 45º 45º 90º Triangle Theorem The length of the hypotenuse in a 45º 45º 90º triangle is 2 times the length of a leg. 190 Chapter Right Triangles
Find the unknown side length in each triangle. Simplify, but do not evaluate the radicals. 6 meters 6 meters 45 45 c a 45 10 inches a 45. Drafting triangles that are 45º 45º 90º triangles are usually described by the length of their hypotenuse. An architect has a 21-centimeter 45º 45º 90º triangle and a 1-centimeter 45º 45º 90º triangle. What are the lengths of the legs of these triangles? Round your answers to the nearest tenth of a centimeter. Lesson. Properties of 45º 45º 90º Triangles 191
4. Drafting triangles usually look like the one shown below with an open triangular area in the center of the triangle. The interior angles of the open triangle have the same measures as the interior angles of the triangle that forms the edges of the drafting triangle. Find the area of the shaded region of the following 45º 45º 90º drafting triangle. Simplify, but do not evaluate any radicals. 1 centimeters 5 centimeters Be prepared to share your solutions and methods. 192 Chapter Right Triangles
.4 Finishing Concrete Properties of 0 o 60 o 90 o Triangles Objectives In this lesson, you will: Find unknown side lengths of 0º 60º 90º triangles. Find areas of 0º 60º 90º triangles. Find volumes and surface areas of solids. Key Term 0º 60º 90º triangle Problem 1 Leveling Off When concrete is poured, the concrete must be leveled off. The tool that is used to level off concrete on a large job, like a section of highway, is called a power screed. Power screeds are gas-powered machines that have bars called screed bars that level off the concrete by vibrating along the concrete s surface. The shape of the screed bars can vary. Some are rectangular prisms, some are right triangular prisms, and some are equilateral triangular prisms. Consider a screed bar in the shape of an equilateral triangular prism. The base is shown on the following grid. One grid square represents a square that is one inch long and one inch wide. Lesson.4 Properties of 0º 60º 90º Triangles 19
A. An equilateral triangle is a regular polygon. What are the measures of the interior angles of this triangle? Take Note Recall that all equilateral triangles are also equiangular. B. Draw the altitude of the equilateral triangle on the diagram on the previous page. Do you know the exact length of the altitude? C. What kinds of triangles are formed within the equilateral triangle by drawing the altitude? How do these triangles compare to each other? D. What are the interior angle measures of these triangles? Which sides of these triangles are the longest? Which sides are the shortest? E. Determine the altitude of the equilateral triangle. Simplify, but do not evaluate any radicals. 194 Chapter Right Triangles
F. The screed bar with the base described in Problem 1 is shown below. Find the surface area and volume of this bar. Round your answers to the nearest tenth of an inch. 72 in 6 in Investigate Problem 1 1. Use the following grid, a compass, a straightedge, and the following instructions to draw an equilateral triangle. First draw a horizontal line segment that represents one side of the triangle. Then take your compass and open the arms so that its ends are at the endpoints of the line segment. Next, place the point of the compass on one of the endpoints and draw an arc. Then place the point of the compass on the other endpoint and draw an arc. The other vertex of the triangle is where the arcs meet. Connect the vertices to form the triangle. Lesson.4 Properties of 0º 60º 90º Triangles 195
2. Draw the altitude of your triangle on the grid following Question 1. Then find the measures of the interior angles and the side lengths of one of the right triangles formed by the altitude. Simplify, but do not evaluate any radicals.. In the right triangle from Problem 1 and the right triangle in Question 1, how does the length of the hypotenuse appear to be related to the length of the shorter leg? Do you think that this relationship is true of any right triangle? Why or why not? 4. Consider the right triangle shown. Complete the following equation, which relates the lengths of the sides of the triangle. 2 2 2 Now write the unknown length by itself on one side of the equation. 2 Finally, write the unknown length as a radical expression in simplest form. How does the length of the longer leg relate to the length of the shorter leg? b 0 60 4 meters 8 meters 196 Chapter Right Triangles
Is this relationship between the legs the same in the right triangles you drew in Question 1? Justify your answer. 5. Just the Math: 0º 60º 90º Triangle The right triangles in this lesson are called 0º 60º 90º triangles. 0º 60º 90º Triangle Theorem The length of the hypotenuse in a 0º 60º 90º triangle is two times the length of the shorter leg, and the length of the longer leg is times the length of the shorter leg. Find the unknown side lengths in each triangle. Do not evaluate the radicals. c 60 meters 0 b b 0 20 feet 60 a Lesson.4 Properties of 0º 60º 90º Triangles 197
a 60 c 8 inches 0 6. Find the area of each triangle. Show all your work and use a complete sentence in your answer. Round your answer to the nearest tenth. 0 18 centimeters 60 5 feet 60 0 198 Chapter Right Triangles
7. Another screed bar is shown below. Find the volume and surface area of the bar. Round your answers to the nearest tenth. 0 4 inches 60 72 inches Be prepared to share your solutions and methods. Lesson.4 Properties of 0º 60º 90º Triangles 199
200 Chapter Right Triangles