homework: collected
DESCRIPTION
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: - PowerPoint PPT PresentationTRANSCRIPT
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Homework: Collected.
1. What do you know about the Pythagorean Theorem?
a) Formula?
b) When and why it’s used?
c) Solve for x:
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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
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Warm-Up:Find the missing angles.
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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.
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Pythagorean Theorem
ca
b
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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
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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
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Why aa22 + b + b22 = c = c22 ? ?
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Proof
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Ex: Find the length of the hypotenuse
a2 + b2 = c2
152 + 202 = x2
225 + 400 = x2
625 = x2
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Ex: Find the length of the leg
a2 + b2 = c2
62 + x2 = 102
36 + x2 = 100 -36 -36 x2 = 64 x = 8
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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
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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
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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
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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
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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:
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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.
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Exit slip: Collected
Page 495: #1 – #6
HW:Pg. 495: #7 – 18, 24 – 32
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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
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Warm-Up: What is Congruent?
1. AB ________
2. BD _______ _______ _______
3. CBE ________ BCE
4. BDE ________
5. ABC ________
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Applying the Pythagorean Theorem
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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.
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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
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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.
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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?
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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?
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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
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Warm-Up: What is Congruent?
1. If AB BC, name two congruent angles. _______ and _______
2. If ACD ADC, name two congruent segments. ______ and ______
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Warm-UpFind the missing angles:
x = ______ y = ______450 900
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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
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Name the missing coordinates of isosceles right triangle ABC.
C(0, 0) A(0, d)
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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
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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
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21. x = 24.8
22. x = 82
23. x = 5.2
24. x = 21.6
25. x = 51
26. x = 17.6
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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