Pythagorean Theorem Pre-Algebra ALCOS 7 Lesson Topics Baseball Definitions Pythagorean...
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Transcript of Pythagorean Theorem Pre-Algebra ALCOS 7 Lesson Topics Baseball Definitions Pythagorean...
Pythagorean Theorem
Pre-Algebra
ALCOS 7
Lesson Topics
• Baseball• Definitions• Pythagorean Theore
m• Converse of the
Pythagorean Theorem
• Application of the Pythagorean Theorem
Baseball A baseball scout uses many
different tests to determine whether or not to draft a particular player. One test for catchers is to see how quickly they can throw a ball from home plate to second base. The scout must know the distance between the two bases in case a player cannot be tested on a baseball diamond. This distance can be found by separating the baseball diamond into two right triangles.
Right Triangles
• Right Triangle – A triangle with one right angle.
• Hypotenuse – Side opposite the right angle and longest side of a right triangle.
• Leg – Either of the two sides that form the right angle.
Leg
Leg
Hypotenuse
Pythagorean Theorem
• In a right triangle, if a and b are the measures of the legs and c is the measure of the hypotenuse, then
a2 + b2 = c2.• This theorem is used to
find the length of any right triangle when the lengths of the other two sides are known.
b
a
c
Finding the Hypotenuse
• Example 1: Find the length of the hypotenuse of a right triangle if a = 3 and
b = 4.4
3c
a2 + b2 = c2
5
5
25
25
169
43
2
2
222
c
c
c
c
c
c
Finding the Length of a Leg
• Example 2: Find the length of the leg of the following right triangle.
9
12
a
a2 + b2 = c2
14481
14481
129
2
2
222
a
a
a
81 81__________________
94.7
63
632
a
a
a
Examples of the Pythagorean Theorem
• Example 3: Find the length of the hypotenuse c when a = 11 and b = 4. Solution
• Example 4: Find the length of the leg of the following right triangle.
Solution
11
4
c
5
13a
Solution of Example 3
• Find the length of the hypotenuse c when
a = 11 and b = 4.
a2 + b2 = c2
11
4
c
70.11
137
137
16121
411
2
2
222
c
c
c
c
c
Solution of Example 4
16925
1352
222
222
a
a
cba • Example 4: Find the length of the leg of the following right triangle.
13a
5
2525_______________
12
144
1442
a
a
a
Converse of the Pythagorean Theorem
• If a2 + b2 = c2, then the triangle with sides a, b, and c is a right triangle.
• If a, b, and c satisfy the equation
a2 + b2 = c2, then a, b, and c are known as Pythagorean triples.
Example of the Converse
Example 5: Determine whether a triangle with lengths 7, 11, and 12 form a right triangle.
**The hypotenuse is the longest length.
14412149
12117?
2?
22
144170
This is not a right triangle.
Example of the Converse
Example 6: Determine whether a triangle with lengths 12, 16, and 20 form a right triangle.
400256144
201612?
2?
22
400400 This is a right triangle. A set of integers such
as 12, 16, and 20 is a Pythagorean triple.
Converse ExamplesExample 7: Determine
whether 4, 5, 6 is a Pythagorean triple.
Example 8: Determine whether 15, 8, and 17 is a Pythagorean triple.
362516
654?
2?
22
36414, 5, and 6 is not a Pythagorean triple.
28964225
17815?
2?
22
289289 15, 8, and 17 is a Pythagorean triple.
Baseball Problem
• On a baseball diamond, the hypotenuse is the length from home plate to second plate. The distance from one base to the next is 90 feet. The Pythagorean theorem can be used to find the distance between home plate to second base.
Solution to Baseball Problem
2
222
222
81008100
9090
c
c
cba
• For the baseball diamond, a = 90 and
b = 90.
90
90
2200,16 cc200,16
127cThe distance from home plate to second base is
approximately 127 feet.
c