14.3 The Pythagorean Theorem - Big Ideas Learning
14.3 The Pythagorean TheoremHow are the lengths of the sides of a righttriangle related?Pythagoras was a Greek mathematician and philosopherwho discovered one of the most famous rules inmathematics. In mathematics, a rule is called a theorem.So, the rule that Pythagoras discovered is called thePythagorean Theorem.Pythagoras(c. 570–c. 490 B.C.)1ACTIVITY: Discovering the Pythagorean TheoremWork with a partner.a. On grid paper, draw anyright triangle. Label thelengths of the two shortersides a and b.b. Label the length of thelongest side c.c2ca2abc. Draw squares along eachof the three sides. Label theareas of the three squaresa 2, b 2, and c 2.Pythagorean TheoremIn this lesson, you will provide geometric proof ofthe Pythagorean Theorem. use the PythagoreanTheorem to find missingside lengths of righttriangles. solve real-life problems.638Chapter 14ms accel pe 1403.indd 638b2d. Cut out the three squares.Make eight copies of the righttriangle and cut them out.Arrange the figures to formtwo identical larger squares.e. MODELING ThePythagorean Theoremdescribes the relationshipamong a 2, b 2, and c 2. Useyour result from part (d) towrite an equation thatdescribes this relationship.a2c2b2Real Numbers and the Pythagorean Theorem2/24/15 9:14:07 AM
2ACTIVITY: Using the Pythagorean Theorem in Two DimensionsWork with a partner. Use a ruler to measure the longest side of eachright triangle. Verify the result of Activity 1 for each right triangle.a.b.2 cm4 cm4.8 cm3 cmc.1d.11in.41in.23 in.3MathPracticeUse DefinitionsHow can you usewhat you knowabout thePythagoreanTheorem to describethe procedure forfinding the length ofthe guy wire?2 in.ACTIVITY: Using the Pythagorean Theorem in Three DimensionsWork with a partner. A guy wire attached 24 feet aboveground level on a telephone pole provides supportfor the pole.a. PROBLEM SOLVING Describe a procedure that youcould use to find the length of the guy wire withoutdirectly measuring the wire.guy wireb. Find the length of the wire when it meets the ground10 feet from the base of the pole.4. IN YOUR OWN WORDS How are the lengths of the sides of a righttriangle related? Give an example using whole numbers.Use what you learned about the Pythagorean Theorem to completeExercises 3 and 4 on page 642.Section 14.3ms accel pe 1403.indd 639The Pythagorean Theorem6392/24/15 9:14:18 AM
14.3 LessonLesson TutorialsKey Vocabularytheorem, p. 638legs, p. 640hypotenuse, p. 640PythagoreanTheorem, p. 640Sides of a Right TriangleThe sides of a right triangle have special names.hypotenuse, cleg, aThe legs are thetwo sides that formthe right angle.Study TipThe hypotenuseis the side oppositethe right angle.leg, bThe Pythagorean TheoremIn a right triangle,the legs are theshorter sides and thehypotenuse is alwaysthe longest side.In any right triangle, the sum of the squares of thelengths of the legs is equal to the square of the lengthof the hypotenuse.WordsAlgebraEXAMPLE1a2 b2 c2Finding the Length of a HypotenuseFind the length of the hypotenuse of the triangle.5mc12 ma2 b2 c2Write the Pythagorean Theorem.52 122 c 2Substitute 5 for a and 12 for b.25 144 c 2Evaluate powers.169 c 2—— 169 c 213 cAdd.Take positive square root of each side.Simplify.The length of the hypotenuse is 13 meters.Find the length of the hypotenuse of the triangle.Exercises 3 and 42.1.c3 in.108 ft2in.5c15 ft640Chapter 14ms accel pe 1403.indd 640Real Numbers and the Pythagorean Theorem2/24/15 9:14:27 AM
2EXAMPLEFinding the Length of a LegFind the missing length of the triangle.a2.1 cma2 b2 c2Write the Pythagorean Theorem.a 2 2.12 2.9 2Substitute 2.1 for b and 2.9 for c.a 2 4.41 8.41Evaluate powers.a2 42.9 cmSubtract 4.41 from each side.a 2Take positive square root of each side.The missing length is 2 centimeters.EXAMPLE3Real-Life ApplicationYou are playing capture the flag. You are 50 yards north and 20 yardseast of your team’s base. The other team’s base is 80 yards north and60 yards east of your base. How far are you from the other team’s base?90NOther Base (60, 80)807060504030 ydYou(20, 50) 40 yd302010WYour BaseStep 1: Draw the situation in a coordinate plane. Let the originrepresent your team’s base. From the descriptions, you areat (20, 50) and the other team’s base is at (60, 80).Step 2: Draw a right triangle with a hypotenuse that represents thedistance between you and the other team’s base. The lengthsof the legs are 30 yards and 40 yards.Step 3: Use the Pythagorean Theorem to find the length of thehypotenuse.10 20 30 40 50 60 70 80 ESa2 b2 c2Write the Pythagorean Theorem.302 402 c 2Substitute 30 for a and 40 for b.900 1600 c 2Evaluate powers.2500 c 2Add.50 cTake positive square root of each side.So, you are 50 yards from the other team’s base.Find the missing length of the triangle.Exercises 5–84.3.a34 yd9.6 m16 yd10.4 mb5. In Example 3, what is the distance between the bases?Section 14.3ms accel pe 1403.indd 641The Pythagorean Theorem6412/24/15 9:14:31 AM
14.3 ExercisesHelp with Homework1. VOCABULARY In a right triangle, how can you tell which sides are the legsand which side is the hypotenuse?2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.Which side is the hypotenuse?Which side is the longest?caWhich side is a leg?bWhich side is opposite the right angle?6) 39 (- 3) 3 (- 9) 4 (- 1)9 (-Find the missing length of the triangle.123.20 km4.5.5.6 in.c7.2 fta21 kmc6. 9 mmb10.6 in.9.6 ft7.8.26 cm12b15 mma4 yd10 cm1yd39. ERROR ANALYSIS Describe and correct the errorin finding the missing length of the triangle. 7 fta2 b2 c225 ft7 2 252 c2674 c2— 674 c5.6 ftc10. TREE SUPPORT How long is the wire that supportsthe tree?3.3 ft642Chapter 14ms accel pe 1403.indd 642Real Numbers and the Pythagorean Theorem2/24/15 9:14:32 AM
Find the missing length of the figure.12.11.5 mm20 cmx13 mmx12 cm35 mmm13. GOLF The figure shows the location of a golf ball aftera tee shot. How many feet from the hole is the ball?xHoleHo14. TENNIS A tennis player asks the referee a question.The sound of the player’s voice travels only 30 feet.Can the referee hear the question? Explain.24 ft180 yd12 ft5 ftHole 13Par 3181 yardsTee15. PROJECT Measure the length, width, and height of arectangular room. Use the Pythagorean Theorem tofind length BC and length AB.16. ALGEBRA The legs of a right triangle have lengthsof 28 meters and 21 meters. The hypotenuse has alength of 5x meters. What is the value of x ?AheightCwidthlengthB17. SNOWBALLS You and a friend stand back-to-back. You run 20 feet forward,then 15 feet to your right. At the same time, your friend runs 16 feet forward,then 12 feet to her right. She stops and hits you with a snowball.a. Draw the situation in a coordinate plane.b. How far does your friend throw the snowball?18.Precision A box has a length of 6 inches, a width of 8 inches, and a heightof 24 inches. Can a cylindrical rod with a length of 63.5 centimeters fit in thebox? Explain your reasoning.Find the square root(s). (Section 14.1)—19. 36—20. 121——21. 16922. 22523. MULTIPLE CHOICE Which equation represents a proportional relationship?(Section 13.3)A y x 1 B y x 1 C y 0.5x Section 14.3ms accel pe 1403.indd 643D y 5 0.5x The Pythagorean Theorem6432/24/15 9:14:36 AM
640 Chapter 14 Real Numbers and the Pythagorean Theorem 14.3 Lesson Lesson Tutorials Key Vocabulary theorem, p. 638 legs, p. 640 hypotenuse, p. 640 Pythagorean Theorem, p. 640 Study Tip In a right triangle, the legs are the shorter sides and the hypotenuse is always the l
Figure 7: Indian proof of Pythagorean Theorem 2.7 Applications of Pythagorean Theorem In this segment we will consider some real life applications to Pythagorean Theorem: The Pythagorean Theorem is a starting place for trigonometry, which leads to methods, for example, for calculating length of a lake. Height of a Building, length of a bridge.File Size: 255KB
Grade 8 Pythagorean Theorem (Relationship) 8.SS.1 Develop and apply the Pythagorean theorem to solve problems. 1. Model and explain the Pythagorean theorem concretely, pictorially, or by using technology. 2. Explain, using examples, that the Pythagorean theorem applies only to right triangles. 3. Determine whether or not a triangle is a right .
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The Pythagorean Theorem gives you a way to find unknown side lengths when you know a triangle is a right triangle. Objectives Use the Pythagorean Theorem and its converse to solve problems. Use Pythagorean inequalities to classify triangles. Vocabulary Pythagorean triple 27-1 The Pythagorean Theorem
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