Do Now: Welcome back! Pass out calculators. Work on practice EOC Week # 16.

Do Now: Welcome back! Pass out calculators. Pick up a transversal line puzzle from the back table and complete.

Triangle Activity: On graph paper, draw a segment that is 3 units long. At one end of this segment, draw a perpendicular segment that is 4 units long. Draw a third segment to form a triangle. Cut out the triangle. Cut out a 3-by-3 square and a 4-by-4 square from the same graph paper. Place the edges of the squares against the corresponding sides of the right triangle. Trace your triangle and squares onto a plane sheet of paper. Cut the two squares into individual squares of strips. Arrange the squares into a large square along the third side of the triangle.

Triangle Activity Continued: What is the area of each of the three squares? What relationship is there between the areas of the small square and the area of the large square? What is the length of the third side of the triangle? Substitute the side lengths of your triangle into an equation. What is the equation? Do you think the relationship is true for triangles that are not right triangles?

Who is Pythagoras?? 1.Some think that Pythagoras actually stole his famous formula from the Babylonians (who had stolen it from the Indians). 2.He discovered square roots by comparing square piles of rocks and figured out the proportional interval of numbers between musical notes by listening to blacksmiths pounding on different sized anvils.

Objective: To use the Pythagorean Theorem and its converse to find the length of a side of a right triangle.

The Pythagorean Theorem and Football? https://www.nbclearn.com/nfl/cuecard/51220

Pythagorean Theorem: The hypotenuse of right triangle is the side opposite the right angle. It is the longest side of a right triangle. The legs are the two sides that form the right angle.

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Pythagorean Theorem: If a triangle is a right triangle, then the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. You can use the Pythagorean Theorem to find the length of any side of a right triangle.

Example 1. Use the Pythagorean Theorem to find the missing measure. c 12 cm 16 cm a2 + b2 = c2 Use the Pythagorean Theorem. 122 + 162 = c2 Substitute for a and b. The length of the hypotenuse is 20 cm.

Use the Pythagorean theorem EXAMPLE 2 Use the Pythagorean theorem Find the unknown length for the triangle shown. SOLUTION a + b = c 2 Pythagorean theorem a + 6 = 7 2 Substitute 6 for b and 7 for c. a + 36 = 49 2 Simplify. a = 13 2 Subtract 36 from each side. a = 13 Take positive square root of each side. The side length a is = 13 ANSWER

GUIDED PRACTICE for Example 1 1. The lengths of the legs of a right triangle are a = 5 and b = 12. Find c. c = 13 ANSWER

EXAMPLE 2 Use the Pythagorean theorem A right triangle has one leg that is 2 inches longer than the other leg. The length of the hypotenuse is 10 inches. Find the unknown lengths. SOLUTION Sketch a right triangle and label the sides with their lengths. Let x be the length of the shorter leg.

Use the Pythagorean theorem EXAMPLE 2 Use the Pythagorean theorem a + b = c 2 Pythagorean theorem x2 + (x + 2)2 + 4 = ( 10)2 Substitute. x2 + x2 + 4x + 4 = 10 Simplify. 2x2 + 4x – 6 = 0 Write in standard form. 2(x – 1)(x + 3) = 0 Factor. x – 1 = 0 or x + 3 = 0 Zero-product property x = 1 or x = – 3 Solve for x. Because length is nonnegative, the solution x = – 3 does not make sense. The legs have lengths of 1 inch and 1 + 2 = 3 inches. ANSWER

EXAMPLE 3 Standardized Test Practice SOLUTION The path of the kicked ball is the hypotenuse of a right triangle. The length of one leg is 12 yards, and the length of the other leg is 40 yards.

Standardized Test Practice EXAMPLE 3 Standardized Test Practice c = a + b 2 Pythagorean theorem c = 12 + 40 2 Substitute 12 for a and 40 for b. c = 1744 2 Simplify. c = 1744 42 Take positive square root of each side. The correct answer is C. ANSWER

GUIDED PRACTICE for Examples 2 and 3 2. A right triangle has one leg that is 3 inches longer than the other leg. The length of the hypotenuse is 15 inches. Find the unknown lengths. ANSWER 9 in. and 12 in. 3. SWIMMING: A rectangular pool is 30 feet wide and 60 feet long. You swim diagonally across the pool. To the nearest foot, how far do you swim? ANSWER 67 feet

Do Now: Pass out calculators. Have your EOC Practice Packet out. Read Lesson 1 and work through the first lesson if you haven’t started yet. You’ll have about 10 minutes to work on this.

Do Now: Pass out calculators. Pick up transversal puzzle # 4 from table and complete.

Objective: To identify Pythagorean triplets.

Determine right triangles EXAMPLE 4 Determine right triangles Tell whether the triangle with the given side lengths is a right triangle. a. 8, 15, 17 b. 5, 8, 9 8 + 15 = 17 2 ? 5 + 8 = 9 ? 2 64 + 225 = 289 ? 25 + 64 = 81 ? 289 = 289 89 = 81 The triangle is a right triangle. ANSWER The triangle is not a right triangle. ANSWER

Use the converse of the Pythagorean theorem EXAMPLE 5 Use the converse of the Pythagorean theorem CONSTRUCTION A construction worker is making sure one corner of the foundation of a house is a right angle. To do this, the worker makes a mark 8 feet from the corner along one wall and another mark 6 feet from the same corner along the other wall. The worker then measures the distance between the two marks and finds the distance to be 10 feet. Is the corner a right angle? SOLUTION 82 + 62 = 102 ? Check to see if a2 + b2 = c2 when a = 8, b = 6, and c =10. 64 +36 = 100 ? Simplify. 100 = 100 Add.

EXAMPLE 5 Use the converse of the Pythagorean theorem ANSWER Because the sides that the construction worker measured form a right triangle, the corner of the foundation is a right angle.

GUIDED PRACTICE for Examples 4 and 5 Tell whether the triangle with the given side lengths is a right triangle. 4. 7, 11, 13 The triangle is not a right triangle ANSWER 5. 15, 36, 39 The triangle is a right triangle ANSWER 6. 15, 112, 113 The triangle is a right triangle ANSWER

GUIDED PRACTICE for Examples 4 and 5 WINDOW DESIGN A window has the shape of a triangle with side lengths of 120 centimeters, 120 centimeters, and 180 centimeters. Is the window a right triangle? Explain. 7. No. 1202 + 1202 ≠ 1802, so it cannot be a right triangle. ANSWER

Exit Ticket: Can you find the missing side of a triangle? Try: Find the missing side of a right triangle with sides: a = 4, c = 9 2. Can you use Algebra to find the missing sides of a right triangle if given a word problem? Try: A right triangle has one leg that is 1 foot longer than the other leg. The hypotenuse is √13 feet. Find the unknown lengths. 3. Can you determine whether a triangle is a right triangle or not? Try: Tell whether the triangle with the given sides is a right triangle: 7, 23, 24