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Homework: Collected.

Homework: Collected. 10. x. 3. 6. 4. x. What do you know about the Pythagorean Theorem? Formula? When and why it’s used? Solve for x :. SWBAT… classify triangles in the coordinate plane. Agenda Notes – 2 slides (20 min) 4 examples (15 min) Exit slip (15 min) Warm-Up:

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Homework: Collected.

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  1. Homework: Collected. 10 x 3 6 4 x • What do you know about the Pythagorean Theorem? • Formula? • When and why it’s used? • Solve for x:

  2. SWBAT… classify triangles in the coordinate plane Agenda Notes – 2 slides (20 min) 4 examples (15 min) Exit slip (15 min) Warm-Up: Write your HW in your planners Set up your Cornell Notes. Topic is “Pythagorean Theorem” Homework: Pg. 495: #7 – 18, 24 – 32 Thurs, 3/13

  3. Warm-Up:Find the missing angles.

  4. Who was he? Greek mathematician named Pythagoras Born ~569 BC on the Greek island of Samos Founded a school for the study of philosophy, mathematics and science. Used mathematics as a means to understand the natural world - First to teach that the earth was a sphere that revolves around the sun Today, the Pythagorean Theorem is one of the most famous theorems in geometry. The relationship it describes has been known for thousands of years.

  5. c a b Pythagorean Theorem • In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. • a2 + b2 = c2 • Side “a” and “b” are called the legs (can be switched around) • Side “c” is called the hypotenuse. • Side “c” must always be the longest side • Side “c” is always opposite the right angle (900)

  6. When do I use the Pythagorean Theorem? If I know the length of any two sides of a right triangle and I need to know the length of third side

  7. The Pythagorean Theorem • “For any right triangle, the sum of the areas of the two small squares is equal to the area of the larger.” • a2 + b2 = c2

  8. Why a2 + b2 = c2 ?

  9. Proof

  10. a2 + b2 = c2 152 + 202 = x2 225 + 400 = x2 625 = x2 Ex: Find the length of the hypotenuse x 15 20

  11. a2+ b2 = c2 62+ x2= 102 36 + x2= 100 -36 -36 x2= 64 x = 8 Ex: Find the length of the leg 10 6 x

  12. a2 + b2 = c2 102 + 242 = x2 100 + 576 = x2 676 = x2 26 = x Ex: The legs of a right triangle have lengths 10 and 24. What is the length of the hypotenuse? x 10 24

  13. Ex: Is the triangle a right triangle?Explain. 28 20 19 a2 + b2 = c2 202 + 192 = 282 400 + 361 = 784 761 = 784 Answer: NO, because a2 + b2 does not equal c2

  14. Pythagorean Triples Whole numbers a, b, and c that satisfy the equationa2 + b2 = c2. Some common Pythagorean Triples: 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25

  15. Ex: Do 16, 48, and 50 form a Pythagorean Triple? a2+ b2 = c2 162+ 482= 502 256 + 2304 = 2500 2560 = 2500 Answer: No, since 16, 48, and 50 did not satisfy a2 + b2 = c2

  16. Determining Type of Triangle: If c2 = a2 + b2 then you know it is a right triangle. If c2 > a2 + b2 then you know it is an obtuse triangle. If c2 < a2 + b2 then you know it is an acute triangle.

  17. Ex. Is the triangle with side lengths 4, acute, right or obtuse? c2 a2 + b2 42 + 16 7 + 11 16 < 18 Answer: Since c2 < a2 + b2, the triangle is acute.

  18. Exit slip: Collected Page 495: #1 – #6 HW: Pg. 495: #7 – 18, 24 – 32

  19. Error Analysis: A triangle has side lengths of 16, 34, and 30. Your friend says it is not a right triangle. Look at your friends work and describe the error. 162+ 342= 302 256 + 1,156 = 900 1,412 = 900

  20. Warm-Up: What is Congruent? • AB  ________ • BD  _______  _______  _______ • CBE  ________ BCE • BDE  ________ • ABC ________

  21. Applying the Pythagorean Theorem

  22. Tim rode 8 miles due north, then 3 miles due east. How far, to the nearest mile, is Tim from where he started? Draw a picture: a2 + b2 = c2 82 + 32 = c2 64 + 9 = c2 73 = c2 3 8 x c = 8.5440037 Tim is 9 miles from where he started.

  23. a2 + b2 = c2 x2 + 52 = 152 x2 + 25 = 225 - 25 -25 x2 = 200 x = 14.142135 The ladder reaches 14.1 feet up the wall. Draw a picture: 15 x A 15 foot ladder leans up against a building. The foot of the ladder is 5 feet from the base of the building. How high up the wall does the ladder reach? 5

  24. Draw a picture and solve: a2 + b2 = c2 32 + 42 = c2 9 + 16 = c2 25 = c2 3 4 x 4 3 5 = c The length of each side of the rhombus is 5 cm. The diagonals of a rhombus are 6 cm and 8 cm. What is the length of each side of the rhombus?

  25. A person can travel from NYC to Buffalo by going north 170 miles to Albany and then west 280 miles to Buffalo. If a highway is built to connect NYC and Buffalo, how many miles would be saved on the trip?

  26. Find length of new highway Old Distance: 280 + 170 = 450 New Distance: 327.566 Saved Miles: 122.4 or 122 miles 280 miles Albany 170 miles ??? New York City a2 + b2 = c2 2802+1702=c2 107300 m= c2 Buf 327.566= c Did I answer question? How many miles would be saved?

  27. B) With gas prices at $3.10 and a vehicle that gets 18 mpg, how much money would be saved roundtrip, if the new highway was traveled instead of the old route? • Saved Miles: 122 miles x 2 = 244 • Cost to drive one mile (gas): • $3.10 divided by 18. ($0.1722…) • Cost to drive 244 miles • $0.1722 times 244 • Saved: $42.02

  28. Warm-Up: What is Congruent? • 1. If AB BC, name two congruent angles. _______ and _______ • 2. If ACD  ADC, name two congruent segments. ______ and ______

  29. Warm-UpFind the missing angles: 450 900 x = ______ y = ______

  30. The slope ofOPis 2 – 0 3 – 0 1 . – 2. The slope ofOQis = = – 2 – 0 2 6 – 0 Using slopes, determine if PQO is a right triangle. Explain. SOLUTION • Check for right angles by checking the slopes. • There is a right angle in the triangle if any of the slopes are perpendicular. Explanation PQOis a right triangle because the slopes of the legs have opposite signs and reciprocals which means they are perpendicular and form a right angle.

  31. Name the missing coordinates of isosceles right triangle ABC. C(0, 0) A(0, d)

  32. Applying the Pythagorean Theorem Answers x = 15 km x = 10 blocks x = 8.5 in x = 8.7 m x = 32.2 ft x = 90.1 ft x = 8.5 ft x = 96 ft x = 101.8 ft x = 24.9 in

  33. Applying the Pythagorean Theorem 11. x = 30 in. No, the box is too small. 12. x = 340 ft 13. x = 8.2 ft 14. x = 8.1 mi 15. Yes, it is a right triangle because a2 + b2 = c2 16. Yes, it is a right triangle because a2 + b2 = c2 17. No, it is not a right triangle because a2 + b2 ≠ c2 18. Yes, it is a triple because a2 + b2 = c2 19. No, it is not a triple because a2 + b2 ≠ c2 20. Yes, it is a triple because a2 + b2 = c2

  34. 21. x = 24.8 22. x = 82 23. x = 5.2 24. x = 21.6 25. x = 51 26. x = 17.6

  35. Pythagorean Theorem Mini-project Project Part One is complete! Create 5 original application problems Labeled diagram Solution with complete sentences Due: Wednesday – beginning of class

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