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a
b
c
222 cba
Pythagorean Theorem Essential Questions
How is the Pythagorean Theorem used to identify side lengths?
When can the Pythagorean Theorem be used to solve real life patterns?
This is a right triangle:
We call it a right triangle because it contains a right angle.
The measure of a right angle is 90o
90o
The little square
90o
in theangle tells you it is aright angle.
About 2,500 years ago, a Greek mathematician named Pythagorus discovered a special relationship between the sides of right triangles.
Pythagorus realized that if you have a right triangle,
3
4
5
and you square the lengths of the two sides that make up the right angle,
24233
4
5
and add them together,
3
4
5
2423 22 43
22 43
you get the same number you would get by squaring the other side.
222 543 3
4
5
Is that correct?
222 543 ?
25169 ?
It is. And it is true for any right triangle.
8
6
10222 1086
1006436
The two sides which come together in a right angle are called
The two sides which come together in a right angle are called
The two sides which come together in a right angle are called
The lengths of the legs are usually called a and b.
a
b
The side across from the right angle
a
b
is called the
And the length of the hypotenuse
is usually labeled c.
a
b
c
The relationship Pythagorus discovered is now called The Pythagorean Theorem:
a
b
c
The Pythagorean Theorem says, given the right triangle with legs a and b and hypotenuse c,
a
b
c
then
a
b
c
.222 cba
You can use The Pythagorean Theorem to solve many kinds of problems.
Suppose you drive directly west for 48 miles,
48
Then turn south and drive for 36 miles.
48
36
How far are you from where you started?
48
36?
482
Using The Pythagorean Theorem,
48
36c
362+ = c2
Why? Can you see that we have a right triangle?
48
36c
482 362+ = c2
Which side is the hypotenuse? Which sides are the legs?
48
36c
482 362+ = c2
22 3648
Then all we need to do is calculate:
12962304
3600 2c
And you end up 60 miles from where you started.
48
3660
So, since c2 is 3600, c is 60.So, since c2 is 3600, c is
Find the length of a diagonal of the rectangle:
15"
8"?
Find the length of a diagonal of the rectangle:
15"
8"?
b = 8
a = 15
c
222 cba 222 815 c 264225 c 2892 c 17c
b = 8
a = 15
c
Find the length of a diagonal of the rectangle:
15"
8"17
Practice using The Pythagorean Theorem to solve these right triangles:
5
12
c = 13
10
b
26
10
b
26
= 24
(a)
(c)
222 cba 222 2610 b
676100 2 b1006762 b
5762 b24b
Check It Out! Example 2
A rectangular field has a length of 100 yards and a width of 33 yards. About how far is it from one corner of the field to the opposite corner of the field? Round your answer to the nearest tenth.
Check It Out! Example 2 Continued
11 Understand the Problem
Rewrite the question as a statement.
• Find the distance from one corner of the field to the opposite corner of the field.
• The segment between the two corners is the hypotenuse.
• The sides of the fields are legs, and they are 33 yards long and 100 yards long.
List the important information:
• Drawing a segment from one corner of the field to the opposite corner of the field divides the field into two right triangles.
Check It Out! Example 2 Continued
22 Make a Plan
You can use the Pythagorean Theorem towrite an equation.
Check It Out! Example 2 Continued
Solve33
a2 + b2 = c2
332 + 1002 = c2
1089 + 10,000 = c2
11,089 = c2
105.304 c
The distance from one corner of the field to the opposite corner is about 105.3 yards.
Use the Pythagorean Theorem.
Substitute for the known variables.
Evaluate the powers.
Add.
Take the square roots of both sides.
105.3 c Round.
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
Proof
a
a
a2
bb
cc
b2
c2
Let’s look at it this way…
Baseball Problem
A baseball “diamond” is really a square.
You can use the Pythagorean theorem to find distances around a baseball diamond.
Baseball Problem
The distance between
consecutive bases is 90
feet. How far does a
catcher have to throw
the ball from home
plate to second base?
Baseball Problem
To use the Pythagorean theorem to solve for x, find the right angle.
Which side is the hypotenuse?
Which sides are the legs?
Now use: aa22 + b + b22 = c = c22
Baseball ProblemSolution
• The hypotenuse is the distance from home to second, or side x in the picture.
• The legs are from home to first and from first to second.
• Solution:
x2 = 902 + 902 = 16,200
x = 127.28 ft
Ladder Problem
A ladder leans against a second-story window of a house. If the ladder is 25 meters long, and the base of the ladder is 7 meters from the house, how high is the window?
Ladder ProblemSolution
• First draw a diagram that shows the sides of the right triangle.
• Label the sides: – Ladder is 25 m
– Distance from house is 7 m
• Use a2 + b2 = c2 to solve for the missing side.
Distance from house: 7 meters
Ladder ProblemSolution
72 + b2 = 252
49 + b2 = 625 b2 = 576 b = 24 m
How did you do? A = 7 m
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