Download calamity lesson #1

Calamity Day Lesson #1
The Pythagorean Theorem and its Converse
Directions: pages 1 & 2 are notes, Do the assignment on pages 3 & 4.
Additional References:
This first website has fantastic examples worked out step by step:
http://www.mathwarehouse.com/geometry/triangles/how-to-use-the-pythagorean-theorem.php
Another good website with examples:
http://www.mathsisfun.com/pythagoras.html
Notes:
The Pythagorean Theorem
This theorem allows us to find an unknown side length on a
right triangle
Note the parts of a right triangle:
hypotenuse
leg
leg
leg2+leg2=hypotenuse2
The pythagorean theorem is:
Example:
5
x
Solution :
52+92 = x2
25 + 81 = x2
106 = x2
106 = x
LEAVE ANSWERS IN RADICAL (SQUARE ROOT) FORM!!
9
Pythagorean Triples
Three whole numbers that fit the pythagorean theorem.
Example
Solve for y. Tell whether the side lengths form a Pythagorean triple
Solution:
y2+ 92=122
y2+81 = 144
y2 = 63
is 63 a whole number?? No…
form a Pythagorean Triple.
Example 2
y=
63
63 = 7.94 If any of the side lengths have a decimal, they do not
Solve for x. Tell whether the side lengths form a Pythagorean Triple.
Solution:
72+x2 = 252
49 + x2 = 625
25
x2 = 576
x = 576 = 24
So the three side lengths are 7, 24, and 25, and these form a Pythagorean Triple.
*Finding the area of a figure using the Pythagorean Theorem: Area is ½ b*h, where the base and height are
the legs of the right triangle. If only one is given, you’ll need to use the Pythagorean Theorem to find the
other before you can use the area formula.
A
Example: Find the area of the right triangle below.
8
15
Area = ½ b*h -or½ leg * leg
one leg = 8. To find the other, use Pythagorean Thm.
82+x2=152
C
64+x2=225
B
2
x =161
x=12.69
---------- now use area formula: ½ * 8 * 12.69 = 50.75  final answer
*Another example: The variables r and s represent the lengths of the legs of a right triangle, and t represents
the length of the hypotenuse. The values of r, s, and t form a Pythagorean Triple. Find the unknown value if
r = 11 and t = 60
Set up the Pythagorean Theorem as rs+s2=t2 and plug in the given values.
112+s2 = 612
Then solve like in the examples above.
121 +s2 = 3721
s2 = 3600
s =60
Section 9.3 Converse of the Pythagorean Theorem
Used to determine what type of triangle is formed by three given sides.
Steps:
1. First make sure that the three sides form a triangle
-The two smallest sides must add up to be larger than the third side
For example: 4, 6, and 9 could form the sides of a triangle because 4 + 6 > 9
2, 9, and 3 could not form the sides of a triangle because 2 + 3 < 9
5, 8, and 3 could not form the sides of a triangle because 5 + 3 = 8
…If the numbers form a triangle, proceed to step 2.
Example:
4, 6, and 9
2. Square the two smaller numbers and add them together.
42+62 = 52
3. Square the largest number
92 = 81
4. Compare (let the two smaller numbers be a & b, and the largest number be c)
If: a2 + b2 = c2 right
a2 + b2 > c2 acute
a2 + b2 < c2 obtuse
In the example above, 52 < 81, so this is an obtuse triangle.
Example 2: 4, 8, 9
4+8 > 9, so these numbers can form the side lengths of a triangle
42 + 82
92
16 + 64
81
80
<
81
so this is an obtuse triangle.
Assignment:
For questions 1-9, use the Pythagorean Theorem to solve for x. Tell whether the sides form a Pythagorean
Triple.
1.
2. x
23
3.
x
53
15
x
18
20
45
4.
5.
6.
x
x
8
21
x
34
15
20
9
7.
8.
9.
x
x
x
9
44
54
13
12
9
10. The variables r and s represent the lengths of the legs of a right triangle, and t represents the length of the
hypotenuse. The values of r, s, and t form a Pythagorean Triple. Find the unknown value if s = 25 and t = 32
11. The variables r and s represent the lengths of the legs of a right triangle, and t represents the length of the
hypotenuse. The values of r, s, and t form a Pythagorean Triple. Find the unknown value if r = 17 and s = 59
For questions 12-13, find the area of the given right triangle.
12.
13.
12
5
37
4
For questions 14-22, determine if the given values can form the side lengths of a triangle. If so, state whether
this triangle is acute, obtuse, or right.
14.
28, 34, 49
15.
3, 9, 10
16. 65, 72, 97
17.
7, 2, 12
18.
77, 36, 85
19.
4, 5, 5
20.
2, 5, 6
21.
8, 6, 2
22.
45, 55, 65