Homework: Collected. 1. What do you know about the Pythagorean Theorem? a) Formula? b) When and why it’s used? c) Solve for x:

SWBAT… classify triangles in the coordinate plane Agenda 1. Notes – 2 slides (20 min) 2. 4 examples (15 min) 3. Exit slip (15 min) Warm-Up: 1. Write your HW in your planners 2. Set up your Cornell Notes. Topic is “Pythagorean Theorem” Homework: Pg. 495: #7 – 18, 24 – 32 Thurs, 3/13

Warm-Up: Find the missing angles.

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.

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. a 2 + b 2 = c 2  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 (90 0 ) c a b

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

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.” a 2 + b 2 = c 2 a 2 + b 2 = c 2

a 2 + b 2 = c 2 ? Why a 2 + b 2 = c 2 ?

Proof

Ex: Find the length of the hypotenuse a 2 + b 2 = c = x = x = x 2

Ex: Find the length of the leg a 2 + b 2 = c x 2 = x 2 = x 2 = 64 x = 8

Ex: The legs of a right triangle have lengths 10 and 24. What is the length of the hypotenuse? a 2 + b 2 = c = x = x = x 2 26 = x

Ex: Is the triangle a right triangle? Explain. a 2 + b 2 = c = = = 784 Answer: NO, because a 2 + b 2 does not equal c 2

Whole numbers a, b, and c that satisfy the equation a 2 + b 2 = c 2. Some common Pythagorean Triples: 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25 Pythagorean Triples

Ex: Do 16, 48, and 50 form a Pythagorean Triple? a 2 + b 2 = c = = = 2500 Answer: No, since 16, 48, and 50 did not satisfy a 2 + b 2 = c 2

If c 2 = a 2 + b 2 then you know it is a right triangle. If c 2 > a 2 + b 2 then you know it is an obtuse triangle. If c 2 < a 2 + b 2 then you know it is an acute triangle. Determining Type of Triangle:

Ex. Is the triangle with side lengths 4, acute, right or obtuse? c 2 a 2 + b < 18 Answer: Since c 2 < a 2 + b 2, the triangle is acute.

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

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 = ,156 = 900 1,412 = 900

Warm-Up: What is Congruent? 1. AB  ________ 2. BD  _______  _______  _______ 3.  CBE  ________   BCE 4.  BDE  ________ 5.  ABC  ________

Applying the Pythagorean Theorem

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: a 2 + b 2 = c = c = c 2 73 = c 2 c = Tim is 9 miles from where he started.

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? Draw a picture: a 2 + b 2 = c 2 x = 15 2 x = x 2 = 200 x = The ladder reaches 14.1 feet up the wall.

The diagonals of a rhombus are 6 cm and 8 cm. What is the length of each side of the rhombus? Draw a picture and solve: a 2 + b 2 = c = c = c 2 25 = c 2 5 = c The length of each side of the rhombus is 5 cm.

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?

Find length of new highway Old Distance: = 450 New Distance: Saved Miles: or 122 miles Buf Alb any New York City 170 mile s 280 mile s ?????? a 2 + b 2 = c =c m= c = c Did I answer question? How many miles would be saved?

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  $ times 244 Saved: $42.02

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

Warm-Up Find the missing angles:  x = ______  y = ______

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

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

Applying the Pythagorean Theorem Answers 1. x = 15 km 2. x = 10 blocks 3. x = 8.5 in 4. x = 8.7 m 5. x = 32.2 ft 6. x = 90.1 ft 7. x = 8.5 ft 8. x = 96 ft 9. x = ft 10. x = 24.9 in

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 a 2 + b 2 = c Yes, it is a right triangle because a 2 + b 2 = c No, it is not a right triangle because a 2 + b 2 ≠ c Yes, it is a triple because a 2 + b 2 = c No, it is not a triple because a 2 + b 2 ≠ c Yes, it is a triple because a 2 + b 2 = c 2

21. x = x = x = x = x = x = 17.6

Pythagorean Theorem Mini-project Project Part One is complete! 1. Create 5 original application problems 2. Labeled diagram 3. Solution with complete sentences Due: Wednesday – beginning of class