Download 19.2 Pythagorean Theorem

19.2 Pythagorean Theorem
A right triangle is a triangle that
has a right (90 degree) angle. The
2 sides that form the right angle are
called the legs of the triangle and
the side opposite the right angle is
called the hypotenuse.
hypotenuse
leg
leg
leg
leg
hypotenuse
The Pythagorean Theorem relates
the sides of any right triangle. If a
and b represent the legs and c the
hypotenuse, then
c  a  b or c  a  b
2
2
2
2
2
Find the length of the hypotenuse:
24 dm
c
10 dm
Find the length of the hypotenuse:
8.00 cm
c
3.90 cm
Find the length of the missing side
25.0 m
18.0 m
b
Find the length of the missing side
42,600 ft
37,800 ft
b
•How long must a guy wire be to reach
from the top of a 15-m telephone pole to a
point on the ground 10 m from the foot of
the pole?
•Use the Pythagorean theorem.
a b  c
2
2
2
15  10  c
2
2
2
225  100  c
325  c
2
2
325  c
18.028  c
15 m
c
10 m
There are two special right
triangles that we will further
explore. They are called a 45, 45,
90 right triangle and a 30, 60 ,90
right triangle. The numbers refer to
the measure of the angles in
degrees.
The 45, 45, 90 triangle is an
isosceles triangle which means the
two legs have equal measure.
leg
45
leg
45
hypotenuse
Since the legs are equal, by the Pyth.Theorem,
the hypotenuse has length = leg  2
Find the hypotenuse of an
isosceles triangle that has equal
sides of 2 m.
2m
2m
?
Hypotenuse has length =
2 2
 2.828
If the hypotenuse of a 45, 45, 90
triangle is 5 cm long, how long is
each leg.
?
5 cm
Since hyp = leg 2
5  leg 2
5
 leg
2
3.536  leg
An equilateral triangle has 3 equal
sides and 3 equal angles that each
measures 60 degrees. The height
bisects both the angle and opposite
side making a 30, 60, 90 triangle.
30
½ side or ½ hypot
60
60
Applying the Pythagorean Th. to
this 30, 60, 90 triangles gives the
following relations:
30
2x, side opposite right angle or longest side
x 3
other leg
60
x, side opposite 30 or short leg
Find the missing measures.
30
Other leg = short leg 3
Hyp. = 2(short leg) = 2(7 in) = 14 in
?
?
7 3
 12.1in
60
7 in
Find the missing measures.
60
?
?
Other leg = short leg
3
8  Shortleg 3
30
8 cm
Hyp. = 2(short leg) = 2( 4.6in) = 9.2 in
8
 shortleg
3
4.6  shortleg