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
12
b
15
= 9
Support Beam: The skyscrapers are connected by a skywalk with support beams. You can use the Pythagorean Theorem to find the approximate length of each support beam.
Each support beam forms the hypotenuse of a right triangle. The right triangles are congruent, so the support beams are the same length. Use the Pythagorean Theorem to show the length of each support beam (x).
Solution:
(hypotenuse)2 = (leg)2 + (leg)2
x2 = (23.26)2 + (47.57)2
x2 = √ (23.26)2 + (47.57)2
x ≈ 13
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 mDistance from house
is 7 mUse a2 + b2 = c2 to
solve for the missing side. Distance from house: 7 meters