Homework: Collected

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

ca

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.”

aa22 + b + b22 = c = c22

Why aa22 + b + b22 = c = c22 ? ?

Proof

Ex: Find the length of the hypotenuse

a2 + b2 = c2

152 + 202 = x2

225 + 400 = x2

625 = x2

Ex: Find the length of the leg

a2 + b2 = c2

62 + x2 = 102

36 + x2 = 100 -36 -36 x2 = 64 x = 8

Ex: The legs of a right triangle have lengths 10 and 24. What is the length of the hypotenuse? a2 + b2 = c2

102 + 242 = x2

100 + 576 = x2

676 = x2

26 = x

Ex: Is the triangle a right triangle?Explain.

a2 + b2 = c2

202 + 192 = 282

400 + 361 = 784 761 = 784

Answer: NO, because a2 + b2 does not equal c2

Whole numbers a, b, and c that satisfy the equation a2 + b2 = c2.

Some common Pythagorean Triples:

3, 4, 55, 12, 138, 15, 177, 24, 25

Pythagorean Triples

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

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.

Determining Type of Triangle:

Ex. Is the triangle with side lengths 4, acute, right or obtuse?

c2 a2 + b2

42 +

16 7 + 11

16 < 18Answer:

Since c2 < a2 + b2, 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.

162 + 342 = 302

256 + 1,156 = 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: a2 + b2 = c2

82 + 32 = c2

64 + 9 = c2

73 = c2

73 cc = 8.5440037

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: 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.

x 200

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:

a2 + b2 = c2

32 + 42 = c2

9 + 16 = c2

25 = c2

25 c5 = 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: 280 + 170 = 450 New Distance: 327.566 Saved Miles: 122.4 or 122 miles

Buf Albany

New York City

170 mile

s

280 mile

s???

a2 + b2 = c2

2802+1702=c2

107300 m= c2

107300 c2

327.566= cDid 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 = 244Cost to drive one mile (gas):

$3.10 divided by 18. ($0.1722…)Cost to drive 244 miles

$0.1722 times 244Saved: $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-UpFind the missing angles:

x = ______ y = ______450 900

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

=21 .

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 = 101.8 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 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

21. x = 24.8

22. x = 82

23. x = 5.2

24. x = 21.6

25. x = 51

26. 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