Homework: Collected.

1
Homework Collected.

• What do you know about the Pythagorean Theorem?
• Formula?
• When and why its used?
• Solve for x

10
x
3
6
4
x
2
SWBAT classify triangles in the coordinate
plane
Thurs, 3/13

• 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

3
Warm-UpFind 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
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
Ex Find the length of the hypotenuse

• a2 b2 c2
• 152 202 x2
• 225 400 x2
• 625 x2

x
15
 
20
 
11
Ex Find the length of the leg

• a2 b2 c2
• 62 x2 102
• 36 x2 100
• -36 -36
• x2 64
• x 8

10
6
x
12
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

x
10
24
13
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

28
20
19
14
Pythagorean Triples

• Whole numbers a, b, and c that satisfy the
  equation a2 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 gt a2 b2 then you know it is an obtuse
  triangle.
• If c2 lt 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 lt 18
• Answer
• Since c2 lt 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?

1. AB ? ________
2. BD ? _______ ? _______ ? _______
3. ?CBE ? ________ ? ?BCE
4. ?BDE ? ________
5. ?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
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.
15
x
5
24
Draw a picture and solve

• The diagonals of a rhombus are 6 cm and 8 cm.
  What is the length of each side of the rhombus?

a2 b2 c2 32 42 c2 9 16 c2 25 c2
3
4
x
3
4
5 c
The length of each side of the rhombus is 5 cm.
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
a2 b2 c2 28021702c2 107300 m c2
Buf
327.566 c
Did I answer question?

• Old Distance 280 170 450
• New Distance 327.566
• Saved Miles 122.4 or 122 miles

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
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
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.
31
Name the missing coordinates of isosceles right
triangle ?ABC.
C(0, 0) A(0, d)
32
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

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